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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 24640o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
24640.bp4 | 24640o1 | \([0, -1, 0, 639, 50561]\) | \(109902239/4312000\) | \(-1130364928000\) | \([2]\) | \(36864\) | \(0.99266\) | \(\Gamma_0(N)\)-optimal |
24640.bp2 | 24640o2 | \([0, -1, 0, -17281, 842625]\) | \(2177286259681/105875000\) | \(27754496000000\) | \([2]\) | \(73728\) | \(1.3392\) | |
24640.bp3 | 24640o3 | \([0, -1, 0, -5761, -1386879]\) | \(-80677568161/3131816380\) | \(-820986873118720\) | \([2]\) | \(110592\) | \(1.5420\) | |
24640.bp1 | 24640o4 | \([0, -1, 0, -225281, -40856575]\) | \(4823468134087681/30382271150\) | \(7964530088345600\) | \([2]\) | \(221184\) | \(1.8885\) |
Rank
sage: E.rank()
The elliptic curves in class 24640o have rank \(1\).
Complex multiplication
The elliptic curves in class 24640o do not have complex multiplication.Modular form 24640.2.a.o
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.