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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 151725ck
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 151725.cy4 | 151725ck1 | \([1, 0, 1, -28213776, 57679596073]\) | \(6585576176607121/187425\) | \(70687247966015625\) | \([2]\) | \(7077888\) | \(2.7428\) | \(\Gamma_0(N)\)-optimal |
| 151725.cy3 | 151725ck2 | \([1, 0, 1, -28249901, 57524475323]\) | \(6610905152742241/35128130625\) | \(13248557450030478515625\) | \([2, 2]\) | \(14155776\) | \(3.0894\) | |
| 151725.cy5 | 151725ck3 | \([1, 0, 1, -12896776, 119796750323]\) | \(-629004249876241/16074715228425\) | \(-6062571062210299981640625\) | \([2]\) | \(28311552\) | \(3.4360\) | |
| 151725.cy2 | 151725ck4 | \([1, 0, 1, -44181026, -14675383177]\) | \(25288177725059761/14387797265625\) | \(5426350769641168212890625\) | \([2, 2]\) | \(28311552\) | \(3.4360\) | |
| 151725.cy6 | 151725ck5 | \([1, 0, 1, 175061599, -116842446427]\) | \(1573196002879828319/926055908203125\) | \(-349261537220478057861328125\) | \([2]\) | \(56623104\) | \(3.7825\) | |
| 151725.cy1 | 151725ck6 | \([1, 0, 1, -518321651, -4533235539427]\) | \(40832710302042509761/91556816413125\) | \(34530608962377143642578125\) | \([2]\) | \(56623104\) | \(3.7825\) |
Rank
sage: E.rank()
The elliptic curves in class 151725ck have rank \(1\).
Complex multiplication
The elliptic curves in class 151725ck do not have complex multiplication.Modular form 151725.2.a.ck
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
