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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 131859k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 131859.k1 | 131859k1 | \([0, 0, 1, -19266114, 32549124172]\) | \(-9221261135586623488/121324931\) | \(-10405568712462651\) | \([]\) | \(4976640\) | \(2.6318\) | \(\Gamma_0(N)\)-optimal |
| 131859.k2 | 131859k2 | \([0, 0, 1, -18176844, 36391787815]\) | \(-7743965038771437568/2189290237869371\) | \(-187766931445223255379891\) | \([]\) | \(14929920\) | \(3.1811\) |
Rank
sage: E.rank()
The elliptic curves in class 131859k have rank \(2\).
Complex multiplication
The elliptic curves in class 131859k do not have complex multiplication.Modular form 131859.2.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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
