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SageMath
E = EllipticCurve("hk1")
E.isogeny_class()
Elliptic curves in class 273600hk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 273600.hk4 | 273600hk1 | \([0, 0, 0, 17700, 1582000]\) | \(3286064/7695\) | \(-1436071680000000\) | \([2]\) | \(1179648\) | \(1.5924\) | \(\Gamma_0(N)\)-optimal |
| 273600.hk3 | 273600hk2 | \([0, 0, 0, -144300, 17458000]\) | \(445138564/81225\) | \(60634137600000000\) | \([2, 2]\) | \(2359296\) | \(1.9390\) | |
| 273600.hk1 | 273600hk3 | \([0, 0, 0, -2196300, 1252762000]\) | \(784767874322/35625\) | \(53187840000000000\) | \([2]\) | \(4718592\) | \(2.2855\) | |
| 273600.hk2 | 273600hk4 | \([0, 0, 0, -684300, -201782000]\) | \(23735908082/1954815\) | \(2918523156480000000\) | \([2]\) | \(4718592\) | \(2.2855\) |
Rank
sage: E.rank()
The elliptic curves in class 273600hk have rank \(1\).
Complex multiplication
The elliptic curves in class 273600hk do not have complex multiplication.Modular form 273600.2.a.hk
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
