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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 25800a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25800.h3 | 25800a1 | \([0, -1, 0, -134383, 19006012]\) | \(1073544204384256/16125\) | \(4031250000\) | \([4]\) | \(55296\) | \(1.3931\) | \(\Gamma_0(N)\)-optimal |
| 25800.h2 | 25800a2 | \([0, -1, 0, -134508, 18969012]\) | \(67283921459536/260015625\) | \(1040062500000000\) | \([2, 2]\) | \(110592\) | \(1.7397\) | |
| 25800.h4 | 25800a3 | \([0, -1, 0, -72008, 36594012]\) | \(-2580786074884/34615360125\) | \(-553845762000000000\) | \([2]\) | \(221184\) | \(2.0863\) | |
| 25800.h1 | 25800a4 | \([0, -1, 0, -199008, -1025988]\) | \(54477543627364/31494140625\) | \(503906250000000000\) | \([2]\) | \(221184\) | \(2.0863\) |
Rank
sage: E.rank()
The elliptic curves in class 25800a have rank \(1\).
Complex multiplication
The elliptic curves in class 25800a do not have complex multiplication.Modular form 25800.2.a.a
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.
