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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 12495.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12495.e1 | 12495q4 | \([1, 0, 0, -2998850, 1998600225]\) | \(25351269426118370449/27551475\) | \(3241403482275\) | \([2]\) | \(147456\) | \(2.1193\) | |
| 12495.e2 | 12495q3 | \([1, 0, 0, -233780, 14595027]\) | \(12010404962647729/6166198828125\) | \(725447125930078125\) | \([2]\) | \(147456\) | \(2.1193\) | |
| 12495.e3 | 12495q2 | \([1, 0, 0, -187475, 31200000]\) | \(6193921595708449/6452105625\) | \(759083774675625\) | \([2, 2]\) | \(73728\) | \(1.7728\) | |
| 12495.e4 | 12495q1 | \([1, 0, 0, -8870, 729987]\) | \(-656008386769/1581036975\) | \(-186007419071775\) | \([4]\) | \(36864\) | \(1.4262\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12495.e have rank \(1\).
Complex multiplication
The elliptic curves in class 12495.e do not have complex multiplication.Modular form 12495.2.a.e
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
