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SageMath
E = EllipticCurve("cc1")
E.isogeny_class()
Elliptic curves in class 53312.cc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
53312.cc1 | 53312cb4 | \([0, -1, 0, -354433, -56005599]\) | \(159661140625/48275138\) | \(1488852539293564928\) | \([2]\) | \(663552\) | \(2.1922\) | |
53312.cc2 | 53312cb3 | \([0, -1, 0, -323073, -70562911]\) | \(120920208625/19652\) | \(606086928269312\) | \([2]\) | \(331776\) | \(1.8456\) | |
53312.cc3 | 53312cb2 | \([0, -1, 0, -134913, 19114145]\) | \(8805624625/2312\) | \(71304344502272\) | \([2]\) | \(221184\) | \(1.6429\) | |
53312.cc4 | 53312cb1 | \([0, -1, 0, -9473, 222881]\) | \(3048625/1088\) | \(33554985648128\) | \([2]\) | \(110592\) | \(1.2963\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 53312.cc have rank \(0\).
Complex multiplication
The elliptic curves in class 53312.cc do not have complex multiplication.Modular form 53312.2.a.cc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.