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SageMath
E = EllipticCurve("ht1")
E.isogeny_class()
Elliptic curves in class 179520ht
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 179520.p2 | 179520ht1 | \([0, -1, 0, -56321, 2284545]\) | \(75370704203521/35157196800\) | \(9216248197939200\) | \([2]\) | \(1032192\) | \(1.7578\) | \(\Gamma_0(N)\)-optimal |
| 179520.p1 | 179520ht2 | \([0, -1, 0, -752641, 251427841]\) | \(179865548102096641/119964240000\) | \(31447905730560000\) | \([2]\) | \(2064384\) | \(2.1043\) |
Rank
sage: E.rank()
The elliptic curves in class 179520ht have rank \(1\).
Complex multiplication
The elliptic curves in class 179520ht do not have complex multiplication.Modular form 179520.2.a.ht
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.
