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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 2772c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2772.b1 | 2772c1 | \([0, 0, 0, -69, 229]\) | \(-84098304/3773\) | \(-1629936\) | \([3]\) | \(576\) | \(-0.043498\) | \(\Gamma_0(N)\)-optimal |
2772.b2 | 2772c2 | \([0, 0, 0, 351, 621]\) | \(15185664/9317\) | \(-2934184176\) | \([]\) | \(1728\) | \(0.50581\) |
Rank
sage: E.rank()
The elliptic curves in class 2772c have rank \(1\).
Complex multiplication
The elliptic curves in class 2772c do not have complex multiplication.Modular form 2772.2.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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.