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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 67830.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 67830.k1 | 67830j4 | \([1, 1, 0, -1006502, -375717234]\) | \(112763439169361881747561/4410075563770668750\) | \(4410075563770668750\) | \([2]\) | \(1597440\) | \(2.3447\) | |
| 67830.k2 | 67830j2 | \([1, 1, 0, -162752, 17301516]\) | \(476767764597000247561/145575633164062500\) | \(145575633164062500\) | \([2, 2]\) | \(798720\) | \(1.9982\) | |
| 67830.k3 | 67830j1 | \([1, 1, 0, -148172, 21888384]\) | \(359771470011326084041/60079405050000\) | \(60079405050000\) | \([2]\) | \(399360\) | \(1.6516\) | \(\Gamma_0(N)\)-optimal |
| 67830.k4 | 67830j3 | \([1, 1, 0, 447718, 117052314]\) | \(9925122178496507474519/11643791198730468750\) | \(-11643791198730468750\) | \([2]\) | \(1597440\) | \(2.3447\) |
Rank
sage: E.rank()
The elliptic curves in class 67830.k have rank \(1\).
Complex multiplication
The elliptic curves in class 67830.k do not have complex multiplication.Modular form 67830.2.a.k
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 LMFDB 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 LMFDB labels.
