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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 7623p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7623.g6 | 7623p1 | \([1, -1, 1, 1066, -3220]\) | \(103823/63\) | \(-81362482047\) | \([2]\) | \(5120\) | \(0.78274\) | \(\Gamma_0(N)\)-optimal |
| 7623.g5 | 7623p2 | \([1, -1, 1, -4379, -22822]\) | \(7189057/3969\) | \(5125836368961\) | \([2, 2]\) | \(10240\) | \(1.1293\) | |
| 7623.g2 | 7623p3 | \([1, -1, 1, -53384, -4727302]\) | \(13027640977/21609\) | \(27907331342121\) | \([2, 2]\) | \(20480\) | \(1.4759\) | |
| 7623.g3 | 7623p4 | \([1, -1, 1, -42494, 3361790]\) | \(6570725617/45927\) | \(59313249412263\) | \([2]\) | \(20480\) | \(1.4759\) | |
| 7623.g1 | 7623p5 | \([1, -1, 1, -853799, -303442180]\) | \(53297461115137/147\) | \(189845791443\) | \([2]\) | \(40960\) | \(1.8225\) | |
| 7623.g4 | 7623p6 | \([1, -1, 1, -37049, -7687204]\) | \(-4354703137/17294403\) | \(-22335167517477507\) | \([2]\) | \(40960\) | \(1.8225\) |
Rank
sage: E.rank()
The elliptic curves in class 7623p have rank \(0\).
Complex multiplication
The elliptic curves in class 7623p do not have complex multiplication.Modular form 7623.2.a.p
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.
