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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 55770.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55770.h1 | 55770f4 | \([1, 1, 0, -1095968, -442037682]\) | \(30161840495801041/2799263610\) | \(13511510786120490\) | \([2]\) | \(1376256\) | \(2.1334\) | |
| 55770.h2 | 55770f3 | \([1, 1, 0, -403068, 93465858]\) | \(1500376464746641/83599963590\) | \(403521056655884310\) | \([2]\) | \(1376256\) | \(2.1334\) | |
| 55770.h3 | 55770f2 | \([1, 1, 0, -73518, -5860512]\) | \(9104453457841/2226896100\) | \(10748802137544900\) | \([2, 2]\) | \(688128\) | \(1.7868\) | |
| 55770.h4 | 55770f1 | \([1, 1, 0, 10982, -570812]\) | \(30342134159/47190000\) | \(-227777116710000\) | \([2]\) | \(344064\) | \(1.4402\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 55770.h have rank \(0\).
Complex multiplication
The elliptic curves in class 55770.h do not have complex multiplication.Modular form 55770.2.a.h
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.
