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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 3696g
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3696.e4 | 3696g1 | \([0, -1, 0, -2079, -35802]\) | \(62140690757632/6237\) | \(99792\) | \([2]\) | \(1536\) | \(0.39058\) | \(\Gamma_0(N)\)-optimal |
| 3696.e3 | 3696g2 | \([0, -1, 0, -2084, -35616]\) | \(3911877700432/38900169\) | \(9958443264\) | \([2, 2]\) | \(3072\) | \(0.73716\) | |
| 3696.e2 | 3696g3 | \([0, -1, 0, -3704, 29184]\) | \(5489767279588/2847396321\) | \(2915733832704\) | \([2, 4]\) | \(6144\) | \(1.0837\) | |
| 3696.e5 | 3696g4 | \([0, -1, 0, -544, -88592]\) | \(-17418812548/3314597517\) | \(-3394147857408\) | \([2]\) | \(6144\) | \(1.0837\) | |
| 3696.e1 | 3696g5 | \([0, -1, 0, -47264, 3967008]\) | \(5701568801608514/6277868289\) | \(12857074255872\) | \([4]\) | \(12288\) | \(1.4303\) | |
| 3696.e6 | 3696g6 | \([0, -1, 0, 13936, 212640]\) | \(146142660369886/94532266521\) | \(-193602081835008\) | \([4]\) | \(12288\) | \(1.4303\) |
Rank
sage: E.rank()
The elliptic curves in class 3696g have rank \(1\).
Complex multiplication
The elliptic curves in class 3696g do not have complex multiplication.Modular form 3696.2.a.g
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
