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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 12870.n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12870.n1 | 12870w4 | \([1, -1, 0, -210654, 37266210]\) | \(1418098748958579169/8307406250\) | \(6056099156250\) | \([6]\) | \(110592\) | \(1.6423\) | |
12870.n2 | 12870w3 | \([1, -1, 0, -12924, 607068]\) | \(-327495950129089/26547449500\) | \(-19353090685500\) | \([6]\) | \(55296\) | \(1.2957\) | |
12870.n3 | 12870w2 | \([1, -1, 0, -3744, 2808]\) | \(7962857630209/4606058600\) | \(3357816719400\) | \([2]\) | \(36864\) | \(1.0930\) | |
12870.n4 | 12870w1 | \([1, -1, 0, 936, 0]\) | \(124326214271/71980480\) | \(-52473769920\) | \([2]\) | \(18432\) | \(0.74643\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12870.n have rank \(0\).
Complex multiplication
The elliptic curves in class 12870.n do not have complex multiplication.Modular form 12870.2.a.n
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.