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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 39039.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
39039.g1 | 39039j4 | \([1, 1, 1, -1874467, -988571134]\) | \(150902699857302457/59378319\) | \(286607804554071\) | \([2]\) | \(580608\) | \(2.1222\) | |
39039.g2 | 39039j2 | \([1, 1, 1, -117712, -15328864]\) | \(37370253593737/730458729\) | \(3525784767265761\) | \([2, 2]\) | \(290304\) | \(1.7756\) | |
39039.g3 | 39039j1 | \([1, 1, 1, -15467, 375968]\) | \(84778086457/35972937\) | \(173634496068033\) | \([4]\) | \(145152\) | \(1.4290\) | \(\Gamma_0(N)\)-optimal |
39039.g4 | 39039j3 | \([1, 1, 1, 3123, -45150942]\) | \(697864103/182466547263\) | \(-880731172527973767\) | \([2]\) | \(580608\) | \(2.1222\) |
Rank
sage: E.rank()
The elliptic curves in class 39039.g have rank \(0\).
Complex multiplication
The elliptic curves in class 39039.g do not have complex multiplication.Modular form 39039.2.a.g
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.