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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 6936.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6936.p1 | 6936m5 | \([0, 1, 0, -111072, -14285088]\) | \(3065617154/9\) | \(444903671808\) | \([2]\) | \(20480\) | \(1.4644\) | |
6936.p2 | 6936m4 | \([0, 1, 0, -18592, 969488]\) | \(28756228/3\) | \(74150611968\) | \([2]\) | \(10240\) | \(1.1178\) | |
6936.p3 | 6936m3 | \([0, 1, 0, -7032, -218880]\) | \(1556068/81\) | \(2002066523136\) | \([2, 2]\) | \(10240\) | \(1.1178\) | |
6936.p4 | 6936m2 | \([0, 1, 0, -1252, 12320]\) | \(35152/9\) | \(55612958976\) | \([2, 2]\) | \(5120\) | \(0.77125\) | |
6936.p5 | 6936m1 | \([0, 1, 0, 193, 1338]\) | \(2048/3\) | \(-1158603312\) | \([2]\) | \(2560\) | \(0.42468\) | \(\Gamma_0(N)\)-optimal |
6936.p6 | 6936m6 | \([0, 1, 0, 4528, -856992]\) | \(207646/6561\) | \(-324334776748032\) | \([2]\) | \(20480\) | \(1.4644\) |
Rank
sage: E.rank()
The elliptic curves in class 6936.p have rank \(1\).
Complex multiplication
The elliptic curves in class 6936.p do not have complex multiplication.Modular form 6936.2.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.