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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 444360.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
444360.p1 | 444360p5 | \([0, -1, 0, -5544096, 5026252140]\) | \(62161150998242/1607445\) | \(487341157567703040\) | \([2]\) | \(12976128\) | \(2.5007\) | \(\Gamma_0(N)\)-optimal* |
444360.p2 | 444360p3 | \([0, -1, 0, -359896, 72230620]\) | \(34008619684/4862025\) | \(737028293852390400\) | \([2, 2]\) | \(6488064\) | \(2.1541\) | \(\Gamma_0(N)\)-optimal* |
444360.p3 | 444360p2 | \([0, -1, 0, -95396, -10187580]\) | \(2533446736/275625\) | \(10445412327840000\) | \([2, 2]\) | \(3244032\) | \(1.8075\) | \(\Gamma_0(N)\)-optimal* |
444360.p4 | 444360p1 | \([0, -1, 0, -92751, -10841424]\) | \(37256083456/525\) | \(1243501467600\) | \([2]\) | \(1622016\) | \(1.4609\) | \(\Gamma_0(N)\)-optimal* |
444360.p5 | 444360p4 | \([0, -1, 0, 126784, -50802084]\) | \(1486779836/8203125\) | \(-1243501467600000000\) | \([2]\) | \(6488064\) | \(2.1541\) | |
444360.p6 | 444360p6 | \([0, -1, 0, 592304, 388741900]\) | \(75798394558/259416045\) | \(-78649108157315082240\) | \([2]\) | \(12976128\) | \(2.5007\) |
Rank
sage: E.rank()
The elliptic curves in class 444360.p have rank \(2\).
Complex multiplication
The elliptic curves in class 444360.p do not have complex multiplication.Modular form 444360.2.a.p
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.