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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 151725.t
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
151725.t1 | 151725bb3 | \([1, 1, 1, -2149588, 1184300906]\) | \(2912566550041/76531875\) | \(28863959586123046875\) | \([2]\) | \(3538944\) | \(2.5155\) | |
151725.t2 | 151725bb2 | \([1, 1, 1, -307213, -39036094]\) | \(8502154921/3186225\) | \(1201683215422265625\) | \([2, 2]\) | \(1769472\) | \(2.1689\) | |
151725.t3 | 151725bb1 | \([1, 1, 1, -271088, -54425344]\) | \(5841725401/1785\) | \(673211885390625\) | \([2]\) | \(884736\) | \(1.8223\) | \(\Gamma_0(N)\)-optimal |
151725.t4 | 151725bb4 | \([1, 1, 1, 957162, -276738594]\) | \(257138126279/236782035\) | \(-89302229808951796875\) | \([2]\) | \(3538944\) | \(2.5155\) |
Rank
sage: E.rank()
The elliptic curves in class 151725.t have rank \(2\).
Complex multiplication
The elliptic curves in class 151725.t do not have complex multiplication.Modular form 151725.2.a.t
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.