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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 53550.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
53550.c1 | 53550x2 | \([1, -1, 0, -71442, 7366216]\) | \(3540302642521/849660\) | \(9678158437500\) | \([2]\) | \(196608\) | \(1.4810\) | |
53550.c2 | 53550x1 | \([1, -1, 0, -3942, 143716]\) | \(-594823321/428400\) | \(-4879743750000\) | \([2]\) | \(98304\) | \(1.1344\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 53550.c have rank \(2\).
Complex multiplication
The elliptic curves in class 53550.c do not have complex multiplication.Modular form 53550.2.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.