Show commands:
SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 11271h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 11271.e4 | 11271h1 | \([1, 0, 0, -155777, 23651808]\) | \(17319700013617/25857\) | \(624125121633\) | \([4]\) | \(36864\) | \(1.5320\) | \(\Gamma_0(N)\)-optimal |
| 11271.e3 | 11271h2 | \([1, 0, 0, -157222, 23190275]\) | \(17806161424897/668584449\) | \(16138003270064481\) | \([2, 2]\) | \(73728\) | \(1.8786\) | |
| 11271.e2 | 11271h3 | \([1, 0, 0, -401427, -66432960]\) | \(296380748763217/92608836489\) | \(2235352180762955241\) | \([2, 2]\) | \(147456\) | \(2.2252\) | |
| 11271.e5 | 11271h4 | \([1, 0, 0, 63863, 83281178]\) | \(1193377118543/124806800313\) | \(-3012532754224259097\) | \([2]\) | \(147456\) | \(2.2252\) | |
| 11271.e1 | 11271h5 | \([1, 0, 0, -5830292, -5418208077]\) | \(908031902324522977/161726530797\) | \(3903685296243212493\) | \([2]\) | \(294912\) | \(2.5718\) | |
| 11271.e6 | 11271h6 | \([1, 0, 0, 1120158, -449568063]\) | \(6439735268725823/7345472585373\) | \(-177301851367049178237\) | \([2]\) | \(294912\) | \(2.5718\) |
Rank
sage: E.rank()
The elliptic curves in class 11271h have rank \(1\).
Complex multiplication
The elliptic curves in class 11271h do not have complex multiplication.Modular form 11271.2.a.h
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.
