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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 27720bn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 27720.be3 | 27720bn1 | \([0, 0, 0, -642, -5339]\) | \(2508888064/396165\) | \(4620868560\) | \([2]\) | \(12288\) | \(0.57725\) | \(\Gamma_0(N)\)-optimal |
| 27720.be2 | 27720bn2 | \([0, 0, 0, -2847, 53314]\) | \(13674725584/1334025\) | \(248961081600\) | \([2, 2]\) | \(24576\) | \(0.92383\) | |
| 27720.be4 | 27720bn3 | \([0, 0, 0, 3453, 256174]\) | \(6099383804/41507235\) | \(-30984984898560\) | \([2]\) | \(49152\) | \(1.2704\) | |
| 27720.be1 | 27720bn4 | \([0, 0, 0, -44427, 3604246]\) | \(12990838708516/144375\) | \(107775360000\) | \([2]\) | \(49152\) | \(1.2704\) |
Rank
sage: E.rank()
The elliptic curves in class 27720bn have rank \(1\).
Complex multiplication
The elliptic curves in class 27720bn do not have complex multiplication.Modular form 27720.2.a.bn
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
