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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 75.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 75.b1 | 75b7 | \([1, 0, 1, -54001, -4834477]\) | \(1114544804970241/405\) | \(6328125\) | \([2]\) | \(96\) | \(1.0956\) | |
| 75.b2 | 75b5 | \([1, 0, 1, -3376, -75727]\) | \(272223782641/164025\) | \(2562890625\) | \([2, 2]\) | \(48\) | \(0.74901\) | |
| 75.b3 | 75b8 | \([1, 0, 1, -2751, -104477]\) | \(-147281603041/215233605\) | \(-3363025078125\) | \([4]\) | \(96\) | \(1.0956\) | |
| 75.b4 | 75b4 | \([1, 0, 1, -2001, 34273]\) | \(56667352321/15\) | \(234375\) | \([2]\) | \(24\) | \(0.40244\) | |
| 75.b5 | 75b3 | \([1, 0, 1, -251, -727]\) | \(111284641/50625\) | \(791015625\) | \([2, 2]\) | \(24\) | \(0.40244\) | |
| 75.b6 | 75b2 | \([1, 0, 1, -126, 523]\) | \(13997521/225\) | \(3515625\) | \([2, 2]\) | \(12\) | \(0.055868\) | |
| 75.b7 | 75b1 | \([1, 0, 1, -1, 23]\) | \(-1/15\) | \(-234375\) | \([2]\) | \(6\) | \(-0.29071\) | \(\Gamma_0(N)\)-optimal |
| 75.b8 | 75b6 | \([1, 0, 1, 874, -5227]\) | \(4733169839/3515625\) | \(-54931640625\) | \([2]\) | \(48\) | \(0.74901\) |
Rank
sage: E.rank()
The elliptic curves in class 75.b have rank \(0\).
Complex multiplication
The elliptic curves in class 75.b do not have complex multiplication.Modular form 75.2.a.b
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
