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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 1575g
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1575.c6 | 1575g1 | \([1, -1, 1, 220, 222]\) | \(103823/63\) | \(-717609375\) | \([2]\) | \(512\) | \(0.38851\) | \(\Gamma_0(N)\)-optimal |
| 1575.c5 | 1575g2 | \([1, -1, 1, -905, 2472]\) | \(7189057/3969\) | \(45209390625\) | \([2, 2]\) | \(1024\) | \(0.73508\) | |
| 1575.c3 | 1575g3 | \([1, -1, 1, -8780, -312528]\) | \(6570725617/45927\) | \(523137234375\) | \([2]\) | \(2048\) | \(1.0817\) | |
| 1575.c2 | 1575g4 | \([1, -1, 1, -11030, 447972]\) | \(13027640977/21609\) | \(246140015625\) | \([2, 2]\) | \(2048\) | \(1.0817\) | |
| 1575.c1 | 1575g5 | \([1, -1, 1, -176405, 28561722]\) | \(53297461115137/147\) | \(1674421875\) | \([2]\) | \(4096\) | \(1.4282\) | |
| 1575.c4 | 1575g6 | \([1, -1, 1, -7655, 724722]\) | \(-4354703137/17294403\) | \(-196994059171875\) | \([2]\) | \(4096\) | \(1.4282\) |
Rank
sage: E.rank()
The elliptic curves in class 1575g have rank \(1\).
Complex multiplication
The elliptic curves in class 1575g do not have complex multiplication.Modular form 1575.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
