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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 12870.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12870.p1 | 12870f2 | \([1, -1, 0, -202569, -35041267]\) | \(46703838741180867/148720000\) | \(2927255760000\) | \([2]\) | \(64512\) | \(1.6167\) | |
| 12870.p2 | 12870f1 | \([1, -1, 0, -12489, -560755]\) | \(-10945484159427/644300800\) | \(-12681772646400\) | \([2]\) | \(32256\) | \(1.2701\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12870.p have rank \(0\).
Complex multiplication
The elliptic curves in class 12870.p do not have complex multiplication.Modular form 12870.2.a.p
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.
