Show commands:
SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 31878.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31878.p1 | 31878u2 | \([1, -1, 0, -558612, -160559550]\) | \(26444015547214434625/46191222\) | \(33673400838\) | \([2]\) | \(172032\) | \(1.7079\) | |
31878.p2 | 31878u1 | \([1, -1, 0, -34902, -2503872]\) | \(-6449916994998625/8532911772\) | \(-6220492681788\) | \([2]\) | \(86016\) | \(1.3613\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 31878.p have rank \(1\).
Complex multiplication
The elliptic curves in class 31878.p do not have complex multiplication.Modular form 31878.2.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.