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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 32368p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 32368.k2 | 32368p1 | \([0, -1, 0, -8188, -226180]\) | \(34000/7\) | \(12500557334272\) | \([]\) | \(58752\) | \(1.2284\) | \(\Gamma_0(N)\)-optimal |
| 32368.k1 | 32368p2 | \([0, -1, 0, -204708, 35697676]\) | \(531250000/343\) | \(612527309379328\) | \([]\) | \(176256\) | \(1.7777\) |
Rank
sage: E.rank()
The elliptic curves in class 32368p have rank \(1\).
Complex multiplication
The elliptic curves in class 32368p do not have complex multiplication.Modular form 32368.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
