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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 12696.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12696.i1 | 12696d2 | \([0, -1, 0, -605352, 181458540]\) | \(80919167474/14283\) | \(4330284242098176\) | \([2]\) | \(101376\) | \(2.0052\) | |
12696.i2 | 12696d1 | \([0, -1, 0, -34032, 3435228]\) | \(-28756228/16767\) | \(-2541688576883712\) | \([2]\) | \(50688\) | \(1.6586\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12696.i have rank \(0\).
Complex multiplication
The elliptic curves in class 12696.i do not have complex multiplication.Modular form 12696.2.a.i
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.