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SageMath
E = EllipticCurve("ek1")
E.isogeny_class()
Elliptic curves in class 161700ek
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 161700.bk2 | 161700ek1 | \([0, -1, 0, -94733, -13013538]\) | \(-3196715008/649539\) | \(-19104403452750000\) | \([2]\) | \(1382400\) | \(1.8465\) | \(\Gamma_0(N)\)-optimal |
| 161700.bk1 | 161700ek2 | \([0, -1, 0, -1583108, -766131288]\) | \(932410994128/29403\) | \(13836934188000000\) | \([2]\) | \(2764800\) | \(2.1930\) |
Rank
sage: E.rank()
The elliptic curves in class 161700ek have rank \(1\).
Complex multiplication
The elliptic curves in class 161700ek do not have complex multiplication.Modular form 161700.2.a.ek
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.
