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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 558.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
558.c1 | 558c3 | \([1, -1, 0, -2976, -61750]\) | \(3999236143617/62\) | \(45198\) | \([2]\) | \(256\) | \(0.44123\) | |
558.c2 | 558c4 | \([1, -1, 0, -276, 134]\) | \(3196010817/1847042\) | \(1346493618\) | \([2]\) | \(256\) | \(0.44123\) | |
558.c3 | 558c2 | \([1, -1, 0, -186, -928]\) | \(979146657/3844\) | \(2802276\) | \([2, 2]\) | \(128\) | \(0.094655\) | |
558.c4 | 558c1 | \([1, -1, 0, -6, -28]\) | \(-35937/496\) | \(-361584\) | \([2]\) | \(64\) | \(-0.25192\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 558.c have rank \(0\).
Complex multiplication
The elliptic curves in class 558.c do not have complex multiplication.Modular form 558.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.