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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 37191.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 37191.b1 | 37191b6 | \([1, 1, 1, -4562097, 3748648518]\) | \(89254274298475942657/17457\) | \(2053798593\) | \([2]\) | \(393216\) | \(2.0881\) | |
| 37191.b2 | 37191b4 | \([1, 1, 1, -285132, 58483116]\) | \(21790813729717297/304746849\) | \(35853162038001\) | \([2, 2]\) | \(196608\) | \(1.7415\) | |
| 37191.b3 | 37191b5 | \([1, 1, 1, -277047, 61966134]\) | \(-19989223566735457/2584262514273\) | \(-304035900541704177\) | \([2]\) | \(393216\) | \(2.0881\) | |
| 37191.b4 | 37191b3 | \([1, 1, 1, -69042, -6084576]\) | \(309368403125137/44372288367\) | \(5220355354089183\) | \([2]\) | \(196608\) | \(1.7415\) | |
| 37191.b5 | 37191b2 | \([1, 1, 1, -18327, 853236]\) | \(5786435182177/627352209\) | \(73807360036641\) | \([2, 2]\) | \(98304\) | \(1.3949\) | |
| 37191.b6 | 37191b1 | \([1, 1, 1, 1518, 67374]\) | \(3288008303/18259263\) | \(-2148184032687\) | \([2]\) | \(49152\) | \(1.0483\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 37191.b have rank \(1\).
Complex multiplication
The elliptic curves in class 37191.b do not have complex multiplication.Modular form 37191.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
