All Questions
39 questions
175
votes
2
answers
66k
views
Estimating the size of solutions of a diophantine equation
A. Is there natural numbers $a,b,c$ such that $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ is equal to an odd natural number ?
(I do not know any such numbers).
B. Suppose that $\frac{a}{b+c} + \...
- 1,881
15
votes
2
answers
1k
views
Determining the Mordell-Weil group of a universal elliptic curve
Let $K$ be a number field and let $K(a,b)$ be the field of rational functions with two indeterminates over $K$. Consider the elliptic curve $E$ over $K(a,b)$ defined by the Weierstrass equation
\begin{...
- 20.8k
12
votes
0
answers
676
views
Kihara-like Z/6Z elliptic curve families
Shoichi Kihara constructed a family of elliptic curves with Mordell–Weil group $\mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}^3$ (generic rank at least 3) in 2006. Kihara's family produces a number of rank 8 ...
- 913
12
votes
0
answers
350
views
Artin representations appearing in Mordell-Weil groups of elliptic curves
Let $E$ be an elliptic curve defined over $\mathbf{Q}$, and let $K$ be a Galois number field. The Galois group $G=\mathrm{Gal}(K/\mathbf{Q})$ acts on the Mordell-Weil group $E(K)$ and thus on the ...
- 20.8k
11
votes
2
answers
679
views
Z/8Z elliptic curve with a missing generator
We are searching for the rank $6$ elliptic curves with the torsion subgroup $\mathbb{Z}/8\mathbb{Z}$ using the families similar to Allan MacLeod's as described in
A. J. MacLeod, A Simple Method for ...
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10
votes
4
answers
1k
views
Possible groups of K-rational points for elliptic curves over arbitrary fields
It is known that the group $C(\Bbb R)$ has at most two connected components, and the connected component of the identity is isomorphic to $U(1)$ as a topological group (trivially) and $C(\Bbb Q)$ is ...
- 3,738
9
votes
3
answers
737
views
Around the diophantine equation $\frac{a}{2b+3c}+\frac{b}{2c+3a}+\frac{c}{2a+3b}=\text{odd integer}$, over positive integers
I am interested to know if a similar theorem that shows this answer of the post
Estimating the size of solutions of a diophantine equation (this MathOverflow, January 5th 2016) is feasible for a ...
8
votes
3
answers
807
views
Smart elliptic curve rational point search given Reg*#Sha
Hi folks,
Let E be a global minimal model of an elliptic curve over QQ, with a
2-torsion point which generates the torsion subgroup, and with
Mordell-Weil rank 1 (under BSD). Let RegSha be equal to
...
8
votes
4
answers
2k
views
Status of $x^3+y^3+z^3=6xyz$
In
Erik Dofs, Solutions of $x^3 + y^3 + z^3 = nxyz$, Acta Arithmetica 73 (1995) pp. 201–213, doi:10.4064/aa-73-3-201-213, EuDML
the author has studied the Diophantine equation
\begin{equation}
x^3+y^...
- 371
8
votes
2
answers
730
views
An elliptic curve for Ramanujan-type cubic identities?
Given the roots $x_i$ of the depressed cubic,
$$x^3+px+q=0$$
with rational coefficients. It can be shown that, in general, one can find rational $u,v$ such that,
$$(u-x_1)^{1/3}+ (u-x_2)^{1/3}+ (u-...
- 12.6k
8
votes
1
answer
206
views
Integral complete 4-partite graphs
For given block sizes $a<b<c<d$, consider the complete 4-partite graph $K_{a,b,c,d }$.
Can such a graph be integral, i.e. have only integer eigenvalues?
It is easy to see that the ...
- 13.4k
7
votes
1
answer
389
views
Why are some solutions of these diophantine equations off the usual patterns?
This is inspired by a recent question about complete multipartite integral graphs. I am wondering if more can be said about tripartite integral graphs with block sizes $a<b<c$. It is easy to see ...
- 13.4k
7
votes
1
answer
454
views
One more generator needed for a Z/6 elliptic curve
I am trying to find the next rank 8 curve with the torsion subgroup Z/6 using Kihara's family as described in https://arxiv.org/pdf/1503.03667.pdf. Meanwhile, I came across a curve generated by $t=629/...
- 913
7
votes
2
answers
605
views
ℤ/18ℤ elliptic curves over cubic fields
I am working on $\mathbb{Z}/18\mathbb{Z}$ elliptic curves over cubic fields. The curves are created using the formulas on p. 584 of
D. Jeon, C. H. Kim, Y. Lee, Families of elliptic curves over cubic ...
- 913
6
votes
1
answer
222
views
rational points and a local perturbation of an elliptic curve
Let $E_{a,b}$ be an elliptic curve defined by the equation $y^{2}=x^3+ax+b$ where $a,b \in \mathbb{Q}$.
Suppose that for $a=a_{0}$ and $b=b_{0}$ the rank of $E_{a_{0},b_{0}}(\mathbb{Q})=1$.
question:...
- 125
6
votes
0
answers
219
views
Ranks of elliptic curves over cubic fields
We are writing a paper on the ranks of elliptic curves over cubic fields. The curves of different torsion subgroups are created by the formulas in Jeon et al. and by our new parametrizations.
D. Jeon,...
- 913
5
votes
0
answers
303
views
2-descent on elliptic curves, and units modulo squares of units
Setup: Let $p$ be a prime, let $f(x) \in \mathbb{Q}_p[x]$ be a separable monic cubic polynomial cutting out the maximal order $\mathcal{O}_{K_f}$ in the etale algebra $K_f := \mathbb{Q}_p[x]/(f(x))$, ...
5
votes
0
answers
184
views
Given a point $P$ on a genus-$1$ curve over $\mathbb{Q}_p$, is there an $R$ such that $2 \mid [P - R]$ and $x(\overline{P}) \neq x(\overline{R})$?
Let $p \in \mathbb{Z}$ be prime, and let $f \in \mathbb{Z}_p[x]$ be a quartic polynomial with nonzero discriminant. Let $C/\mathbb{Q}_p$ be the genus-$1$ curve with affine equation $y^2 = f(x)$. Let $\...
- 53
4
votes
1
answer
411
views
Z2xZ6 elliptic curves with missing generators
By implementing the techniques described in and similar to
A. Dujella, J. C. Peral, Elliptic curves with torsion group Z/8Z or Z/2Z x Z/6Z, arXiv, Number Theory [math.NT] (2013), arXiv:1306.0027v1
A....
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4
votes
1
answer
560
views
What is the complexity of finding an integral point on an elliptic curve?
Let $E$ be an elliptic curve over rational numbers. We know that the set of integral points on $E$ is finite. What is the complexity of finding a point $P\in E(\mathbb{Z})$?
Indeed I'm trying to find ...
- 369
4
votes
1
answer
415
views
3-, 6-, 12-descent for Z2xZ6 elliptic curves
We are trying to write a snippet of Magma code to clarify the steps in the simplified procedure of applying $3$-, $6$-, $12$-descent and hopefully resolve the missing generator of the following $\...
- 913
4
votes
0
answers
313
views
Action of the Picard Scheme of an Elliptic Fibration
Suppose that we have a surface $X$ defined over a field $k$ (I am interested in $k$ being a number field) and an elliptic fibration $f: X \rightarrow \mathbb{P}^1$, i.e. $f$ is proper and almost all ...
- 241
3
votes
1
answer
395
views
Finding $K$-rational points on $X_0(35)$
Let $K=\mathbb{Q}(\sqrt{-2})$. How can I compute the $K$-rational points on the modular curve $X_0(35)$?
Recall that $X_0(35)$ is a hyperelliptic curve of genus 3 and has the simplified affine model:
\...
- 31
3
votes
1
answer
1k
views
How many points are there on an elliptic curve reduced at a bad prime?
Given an elliptic curve $E$ defined over $\mathbb{Z}$, and a prime $p$, I know that Hasse's theorem gives, when $p$ is a good prime, a relation between the number of solutions over $\mathbb{F}_{p^n}$ ...
- 33
3
votes
1
answer
256
views
Rationalizing and minimizing elliptic curve coefficients
I am working on elliptic curves with torsion group $\mathbb{Z}/14\mathbb{Z}$ over quadratic fields. The curves are constructed using the model $E_1=[0,a,0,b,0]$ following the formulas on p. 13 of
L. ...
- 913
3
votes
1
answer
214
views
Distribution of the rank of $y^2=x^4+x+b^2$
For positive integer $b$ define the curve $C_b : y^2=x^4+x+b^2$.
$C_b$ is genus one and has the rational points: $(0,\pm b),(-1,\pm b)$
and one more point from the reciprocal of the polynomial y=0
...
- 25.4k
3
votes
0
answers
309
views
Rational points and Tesla cards
I'm rapidly approaching 300,000 curves in my ongoing search for Mordell curves of rank >=8.
Currently I'm finding that I have a bottleneck in the code that locates rational points on these curves.
...
- 791
2
votes
1
answer
187
views
Is every sufficiently general monic quartic rational square infinitely often?
Let $f(x)=x^4+b_3 x^3+ b_2 x^2+b_1 x + b_0$.
Let $g(x)=x^4 f(1/x)$. Let $C : g(x)=y^2$.
$C$ is birationally equivalent to $f(x)=y^2$.
The constant coefficient of $g(x)$ is $1$ since $f$ is monic
and $(...
- 25.4k
2
votes
1
answer
188
views
Criterion for existence of integral points on an elliptic curve
Is there a criterion for the (presumably infinite) set of $D \in \mathbb{Z}\setminus \{0\}$ such that
$$Dy^2 = x^3-1728$$
has an integral point over $\mathbb{Q}$ with $y \neq 0$? I'd also be ...
- 479
2
votes
1
answer
326
views
$2$-isogenous to a curve in the Tate normal form
It is well-known that an elliptic curve $E$ that has a point of order $2$ and is represented as $E=[0,a,0,b,0]$ has a $2$-isogenous curve $E^\prime=[0,-2a,0,a^2-4b,0]$, see e.g. p. 507 in
A. Dujella, ...
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2
votes
2
answers
541
views
A new simple formula is needed
The following question is related to the families of high rank elliptic curves with torsion subgroup $\mathbb{Z}/6\mathbb{Z}$.
The SageMath/Python code below produces a list of small fractions $a$ for ...
- 913
2
votes
1
answer
269
views
Perfect square quadratic expression
For a given rational $c\ne-1$, I need to find a rational $x\ne20$ with a small denominator such that $(5cx+100)(5cx-64c+36)$ is a perfect square.
I start with
$y^2=(5cx+100)(5cx-64c+36)$
and ...
- 913
2
votes
1
answer
387
views
Question on x coordinates of Mordell Curves where $y^2=x^3+k$ and $k^2 = 1$ mod $24$
In my ongoing search for Mordell curves of rank 8 and above I have currently identified 144,499 curves of a type where $k$ is squarefree and $k^2 = 1$ mod $24$.
In each case the x coordinates are ...
- 791
2
votes
0
answers
136
views
Similar to a $d$-twist but over a cubic field
This question could be related to my old and Duality's newer questions.
I am building a $\mathbb{Z}/9\mathbb{Z}$ elliptic curve $E$ over $\mathbb{Q}$:
$$E: y^2+(t^3-3t^2+1)xy + t^3(t-1)^3y=x^2$$
For $...
- 913
2
votes
0
answers
198
views
Finding rational points via birational map
Let $C$ be an affine curve given by $p_C(x,y)=0$ where $$ p_C=2x^3y + 2xy^3 +x^3 + y^3 + 5x^2y + 5xy^2 + 2x^2 + 2y^2 + 2x^2y^2 + 2xy $$
and let $\overline{C}$ denote the projective closure of $C$. For ...
- 21
0
votes
1
answer
131
views
Primes of the form $p=u^2+1$ and number of points on the elliptic curve $x^3+a x z^2=y^2 z$
Let $p$ be prime of the form $p=u^2+1$. For $a \in \mathbb{F}_p,a \ne 0$,
define
$E_a : x^3+a x z^2=y^2 z$
Let $B= \lfloor 2 \sqrt{p}\rfloor$
Must we have $(\#E_a(\mathbb{F}_p) -p - 1) \in \{2,-2,B,-B\...
- 25.4k
0
votes
1
answer
371
views
Systems of equations for elliptic curves without $3$-torsion
In his YouTube video New rank records for elliptic curves having rational torsion, Noam Elkies uses systems of equations at 6:16 and 8:38 to present $\mathbb{Z}/3\mathbb{Z}$ curves of rank 14 and rank ...
- 913
0
votes
1
answer
544
views
How I can prove or disprove that $\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}=1$ has solutions in rationals? [duplicate]
The motivation of this question is to look if there is such solution in rational number to the identity which mentioned here, I have done many attempts using Wolfram Alpha to find such pairs of ...
user157361
0
votes
0
answers
108
views
Isogeny classes for elliptic curves over quadratic field
Question. Is it possible for an elliptic curve $E$ over quadratic field $K$ to have two separate (yet connected) isogeny classes?
There are two $\mathbb{Z}/14\mathbb{Z}$ elliptic curves, $E_1$ and $...
- 913