Show commands:
SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 80223c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
80223.j4 | 80223c1 | \([1, 1, 0, -65221, -6438296]\) | \(17319700013617/25857\) | \(45807252777\) | \([2]\) | \(163840\) | \(1.3144\) | \(\Gamma_0(N)\)-optimal |
80223.j3 | 80223c2 | \([1, 1, 0, -65826, -6313545]\) | \(17806161424897/668584449\) | \(1184438135054889\) | \([2, 2]\) | \(327680\) | \(1.6610\) | |
80223.j5 | 80223c3 | \([1, 1, 0, 26739, -22549446]\) | \(1193377118543/124806800313\) | \(-221102859969298593\) | \([2]\) | \(655360\) | \(2.0075\) | |
80223.j2 | 80223c4 | \([1, 1, 0, -168071, 17918520]\) | \(296380748763217/92608836489\) | \(164062202979289329\) | \([2, 2]\) | \(655360\) | \(2.0075\) | |
80223.j6 | 80223c5 | \([1, 1, 0, 468994, 122014941]\) | \(6439735268725823/7345472585373\) | \(-13012952758815977253\) | \([2]\) | \(1310720\) | \(2.3541\) | |
80223.j1 | 80223c6 | \([1, 1, 0, -2441056, 1466719159]\) | \(908031902324522977/161726530797\) | \(286508414625264117\) | \([2]\) | \(1310720\) | \(2.3541\) |
Rank
sage: E.rank()
The elliptic curves in class 80223c have rank \(0\).
Complex multiplication
The elliptic curves in class 80223c do not have complex multiplication.Modular form 80223.2.a.c
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.