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SageMath
E = EllipticCurve("ho1")
E.isogeny_class()
Elliptic curves in class 485520.ho
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
485520.ho1 | 485520ho4 | \([0, 1, 0, -3902175200, -93732701601420]\) | \(66464620505913166201729/74880071980801920\) | \(7403204215445803087185838080\) | \([2]\) | \(495452160\) | \(4.2615\) | |
485520.ho2 | 485520ho3 | \([0, 1, 0, -2771699680, 55688867373428]\) | \(23818189767728437646209/232359312482640000\) | \(22972780289401996501647360000\) | \([4]\) | \(495452160\) | \(4.2615\) | \(\Gamma_0(N)\)-optimal* |
485520.ho3 | 485520ho2 | \([0, 1, 0, -306552800, -653543657100]\) | \(32224493437735955329/16782725759385600\) | \(1659265847399413012915814400\) | \([2, 2]\) | \(247726080\) | \(3.9149\) | \(\Gamma_0(N)\)-optimal* |
485520.ho4 | 485520ho1 | \([0, 1, 0, 72245280, -79437287052]\) | \(421792317902132351/271682182840320\) | \(-26860533491215728566599680\) | \([2]\) | \(123863040\) | \(3.5683\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 485520.ho have rank \(1\).
Complex multiplication
The elliptic curves in class 485520.ho do not have complex multiplication.Modular form 485520.2.a.ho
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.