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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 770e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 770.d4 | 770e1 | \([1, -1, 0, -29, -635]\) | \(-2749884201/176619520\) | \(-176619520\) | \([2]\) | \(256\) | \(0.26225\) | \(\Gamma_0(N)\)-optimal |
| 770.d3 | 770e2 | \([1, -1, 0, -1309, -17787]\) | \(248158561089321/1859334400\) | \(1859334400\) | \([2, 2]\) | \(512\) | \(0.60882\) | |
| 770.d1 | 770e3 | \([1, -1, 0, -20909, -1158507]\) | \(1010962818911303721/57392720\) | \(57392720\) | \([2]\) | \(1024\) | \(0.95539\) | |
| 770.d2 | 770e4 | \([1, -1, 0, -2189, 9845]\) | \(1160306142246441/634128110000\) | \(634128110000\) | \([4]\) | \(1024\) | \(0.95539\) |
Rank
sage: E.rank()
The elliptic curves in class 770e have rank \(1\).
Complex multiplication
The elliptic curves in class 770e do not have complex multiplication.Modular form 770.2.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 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.
