Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 15870.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
15870.a1 | 15870d2 | \([1, 1, 0, -128293, 17625463]\) | \(1577505447721/838350\) | \(124105887543150\) | \([2]\) | \(152064\) | \(1.6541\) | |
15870.a2 | 15870d1 | \([1, 1, 0, -6623, 372657]\) | \(-217081801/285660\) | \(-42287932051740\) | \([2]\) | \(76032\) | \(1.3075\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 15870.a have rank \(0\).
Complex multiplication
The elliptic curves in class 15870.a do not have complex multiplication.Modular form 15870.2.a.a
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.