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Consider the elliptic curves -

$ E_{1}: y^{2}+y=x^{3}+x^{2}-769x-8470 $ $ [\text{Cremona}:19a2] $

$ E_{2}: y^{2}+xy+y=x^{3}-86x-2456 $ $ [\text{Cremona}:38a2] $

with both good ordinary reduction and $\mu$-invariant equal to $2$ at $p= 3$. Let $\Lambda = \mathbb{Z}_{p}[[T]] $ and $ K=\mathbb{Q}_{\infty} $ be the cyclotomic $ \mathbb{Z}_{p} $-extension of $ \mathbb{Q} $. Then the Pontrjagin dual $ X_{E}(\mathbb{Q}_{\infty}) $ of $ Sel_{E}(\mathbb{Q}_{\infty})_{p} $ is a finitely generated torsion $ \Lambda $-module and one has a pseudo-isomorphism

$$ X_{E}(\mathbb{Q}_{\infty}) \sim (\bigoplus_{i=1}^{s}\Lambda/f_{i}(T)^{a_{i}}) \bigoplus (\bigoplus_{j=1}^{t}\Lambda/p^{\mu^j_E}) $$

How to compute whether $ \mu^1_E=\mu^2_E=1$ or $\mu^1_E=2$ in the above decomposition for the given two elliptic curves $?$

Consider the elliptic curves -

$ E_{1}: y^{2}+y=x^{3}+x^{2}-769x-8470 $ $ [\text{Cremona}:19a2] $

$ E_{2}: y^{2}+xy+y=x^{3}-86x-2456 $ $ [\text{Cremona}:38a2] $

with both good ordinary reduction and $\mu_{E_i}$-invariants equal to $2$ at $p= 3$ for $i=1,2$. Let $\Lambda = \mathbb{Z}_{p}[[T]] $ and $ K=\mathbb{Q}_{\infty} $ be the cyclotomic $ \mathbb{Z}_{p} $-extension of $ \mathbb{Q} $. Then the Pontrjagin dual $ X_{E}(\mathbb{Q}_{\infty}) $ of $ Sel_{E}(\mathbb{Q}_{\infty})_{p} $ is a finitely generated torsion $ \Lambda $-module and one has a pseudo-isomorphism

$$ X_{E}(\mathbb{Q}_{\infty}) \sim (\bigoplus_{i=1}^{s}\Lambda/f_{i}(T)^{a_{i}}) \bigoplus (\bigoplus_{j=1}^{t}\Lambda/p^{\mu^j_E}) $$ and $$\mu_E=\sum_{j=1}^t \mu^j_E$$

How to compute whether $ \mu^1_E=\mu^2_E=1$ or $\mu^1_E=2$ in the above decomposition for the given two elliptic curves $?$

Consider the two elliptic curves -

$ E_{1}: y^{2}+y=x^{3}+x^{2}-769x-8470 $ $ [\text{Cremona}:19a2] $

$ E_{2}: y^{2}+xy+y=x^{3}-86x-2456 $ $ [\text{Cremona}:38a2] $

with both good ordinary reduction and $\mu$-invariant equal to $2$ at $p= 3$. Let $\Lambda = \mathbb{Z}_{p}[[T]] $ and $ K=\mathbb{Q}_{\infty} $ be the cyclotomic $ \mathbb{Z}_{p} $-extension of $ \mathbb{Q} $. Then the Pontrjagin dual $ X_{E}(\mathbb{Q}_{\infty}) $ of $ Sel_{E}(\mathbb{Q}_{\infty})_{p} $ is a finitely generated torsion $ \Lambda $-module and one has a pseudo-isomorphism

$$ X_{E}(\mathbb{Q}_{\infty}) \sim (\bigoplus_{i=1}^{s}\Lambda/f_{i}(T)^{a_{i}}) \bigoplus (\bigoplus_{j=1}^{t}\Lambda/p^{\mu_j}) $$

How to compute whether $ \mu_1=\mu_2=1$ or $\mu_1=2$ in the above decomposition for the given two elliptic curves $?$

Consider the elliptic curves -

$ E_{1}: y^{2}+y=x^{3}+x^{2}-769x-8470 $ $ [\text{Cremona}:19a2] $

$ E_{2}: y^{2}+xy+y=x^{3}-86x-2456 $ $ [\text{Cremona}:38a2] $

with both good ordinary reduction and $\mu$-invariant equal to $2$ at $p= 3$. Let $\Lambda = \mathbb{Z}_{p}[[T]] $ and $ K=\mathbb{Q}_{\infty} $ be the cyclotomic $ \mathbb{Z}_{p} $-extension of $ \mathbb{Q} $. Then the Pontrjagin dual $ X_{E}(\mathbb{Q}_{\infty}) $ of $ Sel_{E}(\mathbb{Q}_{\infty})_{p} $ is a finitely generated torsion $ \Lambda $-module and one has a pseudo-isomorphism

$$ X_{E}(\mathbb{Q}_{\infty}) \sim (\bigoplus_{i=1}^{s}\Lambda/f_{i}(T)^{a_{i}}) \bigoplus (\bigoplus_{j=1}^{t}\Lambda/p^{\mu^j_E}) $$

How to compute whether $ \mu^1_E=\mu^2_E=1$ or $\mu^1_E=2$ in the above decomposition for the given two elliptic curves $?$

Consider the two elliptic curves -

$ E_{1}: y^{2}+y=x^{3}+x^{2}-769x-8470 $ $ [\text{Cremona}:19a2] $

$ E_{2}: y^{2}+xy+y=x^{3}-86x-2456 $ $ [\text{Cremona}:38a2] $

with both good ordinary reduction and $\mu$-invariant equal to $2$ at $p= 3$. Let $\Lambda = \mathbb{Z}_{p}[[T]] $ and $ K=\mathbb{Q}_{\infty} $ be the cyclotomic $ \mathbb{Z}_{p} $-extension of $ \mathbb{Q} $. Then the Pontrjagin dual $ X_{E}(\mathbb{Q}_{\infty}) $ of $ Sel_{E}(\mathbb{Q}_{\infty})_{p} $ is a finitely generated torsion $ \Lambda $-module and one has a pseudo-isomorphism

$$ X_{E}(\mathbb{Q}_{\infty}) \sim (\bigoplus_{i=1}^{s}\Lambda/(f_{i}(T)^{a_{i}}))\bigoplus(\bigoplus_{j=1}^{t}\Lambda/(p^{\mu_j})) $$

How to compute whether $ \mu_r=\mu_s=1$ or $\mu_r=2$ in the above decomposition for the given two elliptic curves $?$

Consider the two elliptic curves -

$ E_{1}: y^{2}+y=x^{3}+x^{2}-769x-8470 $ $ [\text{Cremona}:19a2] $

$ E_{2}: y^{2}+xy+y=x^{3}-86x-2456 $ $ [\text{Cremona}:38a2] $

with both good ordinary reduction and $\mu$-invariant equal to $2$ at $p= 3$. Let $\Lambda = \mathbb{Z}_{p}[[T]] $ and $ K=\mathbb{Q}_{\infty} $ be the cyclotomic $ \mathbb{Z}_{p} $-extension of $ \mathbb{Q} $. Then the Pontrjagin dual $ X_{E}(\mathbb{Q}_{\infty}) $ of $ Sel_{E}(\mathbb{Q}_{\infty})_{p} $ is a finitely generated torsion $ \Lambda $-module and one has a pseudo-isomorphism

$$ X_{E}(\mathbb{Q}_{\infty}) \sim (\bigoplus_{i=1}^{s}\Lambda/f_{i}(T)^{a_{i}}) \bigoplus (\bigoplus_{j=1}^{t}\Lambda/p^{\mu_j}) $$

How to compute whether $ \mu_1=\mu_2=1$ or $\mu_1=2$ in the above decomposition for the given two elliptic curves $?$