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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 195195.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 195195.k1 | 195195j3 | \([1, 0, 0, -25435771, 28647800726]\) | \(377049455876971757881/144736610099956875\) | \(698615972259962743861875\) | \([2]\) | \(22708224\) | \(3.2744\) | |
| 195195.k2 | 195195j2 | \([1, 0, 0, -11298076, -14299689145]\) | \(33042817838684613961/823326411132225\) | \(3974039331190723820025\) | \([2, 2]\) | \(11354112\) | \(2.9278\) | |
| 195195.k3 | 195195j1 | \([1, 0, 0, -11229631, -14485188784]\) | \(32445917389944971641/20917681785\) | \(100965654698974065\) | \([2]\) | \(5677056\) | \(2.5813\) | \(\Gamma_0(N)\)-optimal |
| 195195.k4 | 195195j4 | \([1, 0, 0, 1744499, -45374928340]\) | \(121639816754787239/184341956658895035\) | \(-889783415478764484993315\) | \([2]\) | \(22708224\) | \(3.2744\) |
Rank
sage: E.rank()
The elliptic curves in class 195195.k have rank \(1\).
Complex multiplication
The elliptic curves in class 195195.k do not have complex multiplication.Modular form 195195.2.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.
