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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 3312.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3312.p1 | 3312b4 | \([0, 0, 0, -26499, 1660322]\) | \(1378334691074/69\) | \(103016448\) | \([2]\) | \(4096\) | \(1.0120\) | |
3312.p2 | 3312b3 | \([0, 0, 0, -2739, -11950]\) | \(1522096994/839523\) | \(1253401122816\) | \([2]\) | \(4096\) | \(1.0120\) | |
3312.p3 | 3312b2 | \([0, 0, 0, -1659, 25850]\) | \(676449508/4761\) | \(3554067456\) | \([2, 2]\) | \(2048\) | \(0.66539\) | |
3312.p4 | 3312b1 | \([0, 0, 0, -39, 902]\) | \(-35152/1863\) | \(-347680512\) | \([2]\) | \(1024\) | \(0.31882\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3312.p have rank \(0\).
Complex multiplication
The elliptic curves in class 3312.p do not have complex multiplication.Modular form 3312.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.