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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 12696.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12696.c1 | 12696n1 | \([0, -1, 0, -199, 1144]\) | \(4499456/27\) | \(5256144\) | \([2]\) | \(2304\) | \(0.12899\) | \(\Gamma_0(N)\)-optimal |
12696.c2 | 12696n2 | \([0, -1, 0, -84, 2340]\) | \(-21296/729\) | \(-2270654208\) | \([2]\) | \(4608\) | \(0.47557\) |
Rank
sage: E.rank()
The elliptic curves in class 12696.c have rank \(1\).
Complex multiplication
The elliptic curves in class 12696.c do not have complex multiplication.Modular form 12696.2.a.c
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.