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SageMath
E = EllipticCurve("hc1")
E.isogeny_class()
Elliptic curves in class 69696hc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 69696.f3 | 69696hc1 | \([0, 0, 0, -3137772, -1559399600]\) | \(10091699281/2737152\) | \(926664395137820393472\) | \([2]\) | \(3686400\) | \(2.7311\) | \(\Gamma_0(N)\)-optimal |
| 69696.f4 | 69696hc2 | \([0, 0, 0, 8013588, -10145946800]\) | \(168105213359/228637728\) | \(-77405435256356059742208\) | \([2]\) | \(7372800\) | \(3.0777\) | |
| 69696.f1 | 69696hc3 | \([0, 0, 0, -701491692, 7151249081680]\) | \(112763292123580561/1932612\) | \(654286912095526060032\) | \([2]\) | \(18432000\) | \(3.5358\) | |
| 69696.f2 | 69696hc4 | \([0, 0, 0, -700794732, 7166168207440]\) | \(-112427521449300721/466873642818\) | \(-158060342219844851241320448\) | \([2]\) | \(36864000\) | \(3.8824\) |
Rank
sage: E.rank()
The elliptic curves in class 69696hc have rank \(0\).
Complex multiplication
The elliptic curves in class 69696hc do not have complex multiplication.Modular form 69696.2.a.hc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
