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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 212415o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
212415.bc4 | 212415o1 | \([1, 0, 0, -55299000, -158283871425]\) | \(6585576176607121/187425\) | \(532242178301041425\) | \([2]\) | \(14155776\) | \(2.9110\) | \(\Gamma_0(N)\)-optimal |
212415.bc3 | 212415o2 | \([1, 0, 0, -55369805, -157858234248]\) | \(6610905152742241/35128130625\) | \(99755490268072689080625\) | \([2, 2]\) | \(28311552\) | \(3.2576\) | |
212415.bc2 | 212415o3 | \([1, 0, 0, -86594810, 40251932475]\) | \(25288177725059761/14387797265625\) | \(40857903468640883141015625\) | \([2, 2]\) | \(56623104\) | \(3.6042\) | |
212415.bc5 | 212415o4 | \([1, 0, 0, -25277680, -328727338423]\) | \(-629004249876241/16074715228425\) | \(-45648347065470693282562425\) | \([2]\) | \(56623104\) | \(3.6042\) | |
212415.bc1 | 212415o5 | \([1, 0, 0, -1015910435, 12438995138100]\) | \(40832710302042509761/91556816413125\) | \(259999463284141348633963125\) | \([2]\) | \(113246208\) | \(3.9508\) | |
212415.bc6 | 212415o6 | \([1, 0, 0, 343120735, 320684297142]\) | \(1573196002879828319/926055908203125\) | \(-2629777317916929473876953125\) | \([2]\) | \(113246208\) | \(3.9508\) |
Rank
sage: E.rank()
The elliptic curves in class 212415o have rank \(0\).
Complex multiplication
The elliptic curves in class 212415o do not have complex multiplication.Modular form 212415.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.