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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 13872.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 13872.p1 | 13872b5 | \([0, -1, 0, -111072, 14285088]\) | \(3065617154/9\) | \(444903671808\) | \([2]\) | \(40960\) | \(1.4644\) | |
| 13872.p2 | 13872b3 | \([0, -1, 0, -18592, -969488]\) | \(28756228/3\) | \(74150611968\) | \([2]\) | \(20480\) | \(1.1178\) | |
| 13872.p3 | 13872b4 | \([0, -1, 0, -7032, 218880]\) | \(1556068/81\) | \(2002066523136\) | \([2, 2]\) | \(20480\) | \(1.1178\) | |
| 13872.p4 | 13872b2 | \([0, -1, 0, -1252, -12320]\) | \(35152/9\) | \(55612958976\) | \([2, 2]\) | \(10240\) | \(0.77125\) | |
| 13872.p5 | 13872b1 | \([0, -1, 0, 193, -1338]\) | \(2048/3\) | \(-1158603312\) | \([2]\) | \(5120\) | \(0.42468\) | \(\Gamma_0(N)\)-optimal |
| 13872.p6 | 13872b6 | \([0, -1, 0, 4528, 856992]\) | \(207646/6561\) | \(-324334776748032\) | \([2]\) | \(40960\) | \(1.4644\) |
Rank
sage: E.rank()
The elliptic curves in class 13872.p have rank \(1\).
Complex multiplication
The elliptic curves in class 13872.p do not have complex multiplication.Modular form 13872.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
