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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 315.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
315.b1 | 315a3 | \([0, 0, 1, -1182, 16362]\) | \(-250523582464/13671875\) | \(-9966796875\) | \([3]\) | \(180\) | \(0.67677\) | |
315.b2 | 315a1 | \([0, 0, 1, -12, -18]\) | \(-262144/35\) | \(-25515\) | \([]\) | \(20\) | \(-0.42184\) | \(\Gamma_0(N)\)-optimal |
315.b3 | 315a2 | \([0, 0, 1, 78, 45]\) | \(71991296/42875\) | \(-31255875\) | \([3]\) | \(60\) | \(0.12746\) |
Rank
sage: E.rank()
The elliptic curves in class 315.b have rank \(0\).
Complex multiplication
The elliptic curves in class 315.b 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 LMFDB numbering.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.