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SageMath
E = EllipticCurve("ch1")
E.isogeny_class()
Elliptic curves in class 40425.ch
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 40425.ch1 | 40425j4 | \([1, 1, 0, -1079250, -431998125]\) | \(75627935783569/396165\) | \(728256501328125\) | \([2]\) | \(442368\) | \(2.0478\) | |
| 40425.ch2 | 40425j2 | \([1, 1, 0, -68625, -6525000]\) | \(19443408769/1334025\) | \(2452292300390625\) | \([2, 2]\) | \(221184\) | \(1.7012\) | |
| 40425.ch3 | 40425j1 | \([1, 1, 0, -13500, 475875]\) | \(148035889/31185\) | \(57326313515625\) | \([2]\) | \(110592\) | \(1.3547\) | \(\Gamma_0(N)\)-optimal |
| 40425.ch4 | 40425j3 | \([1, 1, 0, 60000, -28005375]\) | \(12994449551/192163125\) | \(-353246867080078125\) | \([2]\) | \(442368\) | \(2.0478\) |
Rank
sage: E.rank()
The elliptic curves in class 40425.ch have rank \(0\).
Complex multiplication
The elliptic curves in class 40425.ch do not have complex multiplication.Modular form 40425.2.a.ch
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
