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SageMath
E = EllipticCurve("dx1")
E.isogeny_class()
Elliptic curves in class 31680dx
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31680.de2 | 31680dx1 | \([0, 0, 0, -9507, 340144]\) | \(2036792051776/107421875\) | \(5011875000000\) | \([2]\) | \(46080\) | \(1.1939\) | \(\Gamma_0(N)\)-optimal |
31680.de1 | 31680dx2 | \([0, 0, 0, -150132, 22390144]\) | \(125330290485184/378125\) | \(1129075200000\) | \([2]\) | \(92160\) | \(1.5404\) |
Rank
sage: E.rank()
The elliptic curves in class 31680dx have rank \(1\).
Complex multiplication
The elliptic curves in class 31680dx do not have complex multiplication.Modular form 31680.2.a.dx
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.