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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 112.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
112.b1 | 112b3 | \([0, 0, 0, -299, -1990]\) | \(1443468546/7\) | \(14336\) | \([2]\) | \(16\) | \(-0.0020328\) | |
112.b2 | 112b4 | \([0, 0, 0, -59, 138]\) | \(11090466/2401\) | \(4917248\) | \([4]\) | \(16\) | \(-0.0020328\) | |
112.b3 | 112b2 | \([0, 0, 0, -19, -30]\) | \(740772/49\) | \(50176\) | \([2, 2]\) | \(8\) | \(-0.34861\) | |
112.b4 | 112b1 | \([0, 0, 0, 1, -2]\) | \(432/7\) | \(-1792\) | \([2]\) | \(4\) | \(-0.69518\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 112.b have rank \(0\).
Complex multiplication
The elliptic curves in class 112.b do not have complex multiplication.Modular form 112.2.a.b
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.