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SageMath
E = EllipticCurve("v1")
E.isogeny_class()
Elliptic curves in class 78400v
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
78400.fa4 | 78400v1 | \([0, 0, 0, -3500, -98000]\) | \(-3375\) | \(-1404928000000\) | \([2]\) | \(73728\) | \(1.0453\) | \(\Gamma_0(N)\)-optimal | \(-7\) |
78400.fa3 | 78400v2 | \([0, 0, 0, -59500, -5586000]\) | \(16581375\) | \(1404928000000\) | \([2]\) | \(147456\) | \(1.3919\) | \(-28\) | |
78400.fa2 | 78400v3 | \([0, 0, 0, -171500, 33614000]\) | \(-3375\) | \(-165288374272000000\) | \([2]\) | \(516096\) | \(2.0183\) | \(-7\) | |
78400.fa1 | 78400v4 | \([0, 0, 0, -2915500, 1915998000]\) | \(16581375\) | \(165288374272000000\) | \([2]\) | \(1032192\) | \(2.3648\) | \(-28\) |
Rank
sage: E.rank()
The elliptic curves in class 78400v have rank \(2\).
Complex multiplication
Each elliptic curve in class 78400v has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-7}) \).Modular form 78400.2.a.v
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 7 & 14 \\ 2 & 1 & 14 & 7 \\ 7 & 14 & 1 & 2 \\ 14 & 7 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.