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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 672.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
672.c1 | 672a2 | \([0, -1, 0, -33, 81]\) | \(1000000/63\) | \(258048\) | \([2]\) | \(64\) | \(-0.21054\) | |
672.c2 | 672a1 | \([0, -1, 0, 2, 4]\) | \(8000/147\) | \(-9408\) | \([2]\) | \(32\) | \(-0.55712\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 672.c have rank \(1\).
Complex multiplication
The elliptic curves in class 672.c do not have complex multiplication.Modular form 672.2.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.