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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 12870.w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12870.w1 | 12870x4 | \([1, -1, 0, -1443834, 668110788]\) | \(456612868287073618849/12544848030000\) | \(9145194213870000\) | \([4]\) | \(196608\) | \(2.1660\) | |
12870.w2 | 12870x3 | \([1, -1, 0, -402714, -88828380]\) | \(9908022260084596129/1047363281250000\) | \(763527832031250000\) | \([2]\) | \(196608\) | \(2.1660\) | |
12870.w3 | 12870x2 | \([1, -1, 0, -93834, 9580788]\) | \(125337052492018849/18404100000000\) | \(13416588900000000\) | \([2, 2]\) | \(98304\) | \(1.8195\) | |
12870.w4 | 12870x1 | \([1, -1, 0, 9846, 809460]\) | \(144794100308831/474439680000\) | \(-345866526720000\) | \([2]\) | \(49152\) | \(1.4729\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12870.w have rank \(1\).
Complex multiplication
The elliptic curves in class 12870.w do not have complex multiplication.Modular form 12870.2.a.w
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.