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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 1344b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1344.c3 | 1344b1 | \([0, -1, 0, -29, -51]\) | \(2725888/21\) | \(21504\) | \([2]\) | \(128\) | \(-0.33961\) | \(\Gamma_0(N)\)-optimal |
1344.c2 | 1344b2 | \([0, -1, 0, -49, 49]\) | \(810448/441\) | \(7225344\) | \([2, 2]\) | \(256\) | \(0.0069590\) | |
1344.c1 | 1344b3 | \([0, -1, 0, -609, 5985]\) | \(381775972/567\) | \(37158912\) | \([2]\) | \(512\) | \(0.35353\) | |
1344.c4 | 1344b4 | \([0, -1, 0, 191, 193]\) | \(11696828/7203\) | \(-472055808\) | \([2]\) | \(512\) | \(0.35353\) |
Rank
sage: E.rank()
The elliptic curves in class 1344b have rank \(1\).
Complex multiplication
The elliptic curves in class 1344b do not have complex multiplication.Modular form 1344.2.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 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.