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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 312.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
312.b1 | 312b1 | \([0, -1, 0, -3, 0]\) | \(256000/117\) | \(1872\) | \([2]\) | \(16\) | \(-0.67717\) | \(\Gamma_0(N)\)-optimal |
312.b2 | 312b2 | \([0, -1, 0, 12, -12]\) | \(686000/507\) | \(-129792\) | \([2]\) | \(32\) | \(-0.33060\) |
Rank
sage: E.rank()
The elliptic curves in class 312.b have rank \(1\).
Complex multiplication
The elliptic curves in class 312.b do not have complex multiplication.Modular form 312.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.