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SageMath
sage: E = EllipticCurve("b1")
sage: E.isogeny_class()
Elliptic curves in class 315b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
315.a3 | 315b1 | [1, -1, 1, -23, -34] | [2] | 32 | \(\Gamma_0(N)\)-optimal |
315.a2 | 315b2 | [1, -1, 1, -68, 182] | [2, 2] | 64 | |
315.a1 | 315b3 | [1, -1, 1, -1013, 12656] | [2] | 128 | |
315.a4 | 315b4 | [1, -1, 1, 157, 992] | [2] | 128 |
Rank
sage: E.rank()
The elliptic curves in class 315b have rank \(1\).
Complex multiplication
The elliptic curves in class 315b do not have complex multiplication.Modular form 315.2.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.