Show commands:
SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 18032p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
18032.o2 | 18032p1 | \([0, 0, 0, -6419, 921298]\) | \(-60698457/725788\) | \(-349750199959552\) | \([2]\) | \(55296\) | \(1.4725\) | \(\Gamma_0(N)\)-optimal |
18032.o1 | 18032p2 | \([0, 0, 0, -186739, 30962610]\) | \(1494447319737/5411854\) | \(2607919969263616\) | \([2]\) | \(110592\) | \(1.8191\) |
Rank
sage: E.rank()
The elliptic curves in class 18032p have rank \(1\).
Complex multiplication
The elliptic curves in class 18032p do not have complex multiplication.Modular form 18032.2.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 Cremona 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 Cremona labels.