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SageMath
E = EllipticCurve("et1")
E.isogeny_class()
Elliptic curves in class 117600.et
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 117600.et1 | 117600dg4 | \([0, 1, 0, -102900408, -401800836312]\) | \(128025588102048008/7875\) | \(7411887000000000\) | \([2]\) | \(7077888\) | \(2.9536\) | |
| 117600.et2 | 117600dg3 | \([0, 1, 0, -7203408, -4678351812]\) | \(43919722445768/15380859375\) | \(14476341796875000000000\) | \([2]\) | \(7077888\) | \(2.9536\) | |
| 117600.et3 | 117600dg1 | \([0, 1, 0, -6431658, -6278961312]\) | \(250094631024064/62015625\) | \(7296076265625000000\) | \([2, 2]\) | \(3538944\) | \(2.6071\) | \(\Gamma_0(N)\)-optimal |
| 117600.et4 | 117600dg2 | \([0, 1, 0, -5666033, -7829351937]\) | \(-2671731885376/1969120125\) | \(-14826560869512000000000\) | \([2]\) | \(7077888\) | \(2.9536\) |
Rank
sage: E.rank()
The elliptic curves in class 117600.et have rank \(1\).
Complex multiplication
The elliptic curves in class 117600.et do not have complex multiplication.Modular form 117600.2.a.et
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.
