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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 1248.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1248.b1 | 1248h2 | \([0, -1, 0, -3744, -86940]\) | \(11339065490696/351\) | \(179712\) | \([2]\) | \(768\) | \(0.51272\) | |
1248.b2 | 1248h3 | \([0, -1, 0, -369, 513]\) | \(1360251712/771147\) | \(3158618112\) | \([4]\) | \(768\) | \(0.51272\) | |
1248.b3 | 1248h1 | \([0, -1, 0, -234, -1296]\) | \(22235451328/123201\) | \(7884864\) | \([2, 2]\) | \(384\) | \(0.16614\) | \(\Gamma_0(N)\)-optimal |
1248.b4 | 1248h4 | \([0, -1, 0, -104, -2856]\) | \(-245314376/6908733\) | \(-3537271296\) | \([2]\) | \(768\) | \(0.51272\) |
Rank
sage: E.rank()
The elliptic curves in class 1248.b have rank \(1\).
Complex multiplication
The elliptic curves in class 1248.b do not have complex multiplication.Modular form 1248.2.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.