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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 345576bi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
345576.bi4 | 345576bi1 | \([0, 1, 0, -85503964, -307173846784]\) | \(-152435594466395827792/1646846627220711\) | \(-746877249988031999966976\) | \([2]\) | \(49766400\) | \(3.3968\) | \(\Gamma_0(N)\)-optimal |
345576.bi3 | 345576bi2 | \([0, 1, 0, -1371591184, -19552183006864]\) | \(157304700372188331121828/18069292138401\) | \(32779113727997761496064\) | \([2, 2]\) | \(99532800\) | \(3.7434\) | |
345576.bi2 | 345576bi3 | \([0, 1, 0, -1375119544, -19446535440400]\) | \(79260902459030376659234/842751810121431609\) | \(3057635818476612613471537152\) | \([2]\) | \(199065600\) | \(4.0900\) | |
345576.bi1 | 345576bi4 | \([0, 1, 0, -21945458344, -1251317839422928]\) | \(322159999717985454060440834/4250799\) | \(15422565841385472\) | \([2]\) | \(199065600\) | \(4.0900\) |
Rank
sage: E.rank()
The elliptic curves in class 345576bi have rank \(0\).
Complex multiplication
The elliptic curves in class 345576bi do not have complex multiplication.Modular form 345576.2.a.bi
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.