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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 493680.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
493680.f1 | 493680f3 | \([0, -1, 0, -163564936, 805216798960]\) | \(66692696957462376289/1322972640\) | \(9599904698740899840\) | \([2]\) | \(58982400\) | \(3.1748\) | \(\Gamma_0(N)\)-optimal* |
493680.f2 | 493680f4 | \([0, -1, 0, -15499656, -1742198544]\) | \(56751044592329569/32660264340000\) | \(236993128670964695040000\) | \([2]\) | \(58982400\) | \(3.1748\) | |
493680.f3 | 493680f2 | \([0, -1, 0, -10233736, 12555827440]\) | \(16334668434139489/72511718400\) | \(526167786948290150400\) | \([2, 2]\) | \(29491200\) | \(2.8282\) | \(\Gamma_0(N)\)-optimal* |
493680.f4 | 493680f1 | \([0, -1, 0, -321416, 391428336]\) | \(-506071034209/8823767040\) | \(-64028023034468106240\) | \([2]\) | \(14745600\) | \(2.4816\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 493680.f have rank \(1\).
Complex multiplication
The elliptic curves in class 493680.f do not have complex multiplication.Modular form 493680.2.a.f
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.