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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 12696.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12696.p1 | 12696s2 | \([0, 1, 0, -119224, 15805376]\) | \(15043017316604/243\) | \(3027538944\) | \([2]\) | \(46080\) | \(1.3646\) | |
12696.p2 | 12696s1 | \([0, 1, 0, -7444, 245600]\) | \(-14647977776/59049\) | \(-183922990848\) | \([2]\) | \(23040\) | \(1.0181\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12696.p have rank \(2\).
Complex multiplication
The elliptic curves in class 12696.p do not have complex multiplication.Modular form 12696.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.