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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 56a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
56.a4 | 56a1 | \([0, 0, 0, 1, 2]\) | \(432/7\) | \(-1792\) | \([4]\) | \(2\) | \(-0.69518\) | \(\Gamma_0(N)\)-optimal |
56.a3 | 56a2 | \([0, 0, 0, -19, 30]\) | \(740772/49\) | \(50176\) | \([2, 2]\) | \(4\) | \(-0.34861\) | |
56.a2 | 56a3 | \([0, 0, 0, -59, -138]\) | \(11090466/2401\) | \(4917248\) | \([2]\) | \(8\) | \(-0.0020328\) | |
56.a1 | 56a4 | \([0, 0, 0, -299, 1990]\) | \(1443468546/7\) | \(14336\) | \([2]\) | \(8\) | \(-0.0020328\) |
Rank
sage: E.rank()
The elliptic curves in class 56a have rank \(0\).
Complex multiplication
The elliptic curves in class 56a do not have complex multiplication.Modular form 56.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.