Show commands:
SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 1682.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1682.b1 | 1682d2 | \([1, 0, 1, -124486, -16915536]\) | \(426477625/8\) | \(4001971303688\) | \([]\) | \(8700\) | \(1.5425\) | |
1682.b2 | 1682d1 | \([1, 0, 1, -2541, 10430]\) | \(3625/2\) | \(1000492825922\) | \([3]\) | \(2900\) | \(0.99318\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1682.b have rank \(0\).
Complex multiplication
The elliptic curves in class 1682.b do not have complex multiplication.Modular form 1682.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.