Show commands:
SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 51714.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51714.b1 | 51714k1 | \([1, -1, 0, -3834, -57056]\) | \(1771561/612\) | \(2153471181732\) | \([2]\) | \(122880\) | \(1.0683\) | \(\Gamma_0(N)\)-optimal |
51714.b2 | 51714k2 | \([1, -1, 0, 11376, -406886]\) | \(46268279/46818\) | \(-164740545402498\) | \([2]\) | \(245760\) | \(1.4148\) |
Rank
sage: E.rank()
The elliptic curves in class 51714.b have rank \(1\).
Complex multiplication
The elliptic curves in class 51714.b do not have complex multiplication.Modular form 51714.2.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.