Show commands:
SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 570f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 570.f3 | 570f1 | \([1, 0, 1, -23, 506]\) | \(-1263214441/110808000\) | \(-110808000\) | \([6]\) | \(144\) | \(0.22322\) | \(\Gamma_0(N)\)-optimal |
| 570.f2 | 570f2 | \([1, 0, 1, -1103, 13898]\) | \(148212258825961/1218375000\) | \(1218375000\) | \([6]\) | \(288\) | \(0.56979\) | |
| 570.f4 | 570f3 | \([1, 0, 1, 202, -13624]\) | \(918046641959/80912056320\) | \(-80912056320\) | \([2]\) | \(432\) | \(0.77253\) | |
| 570.f1 | 570f4 | \([1, 0, 1, -7478, -240952]\) | \(46237740924063961/1806561830400\) | \(1806561830400\) | \([2]\) | \(864\) | \(1.1191\) |
Rank
sage: E.rank()
The elliptic curves in class 570f have rank \(0\).
Complex multiplication
The elliptic curves in class 570f do not have complex multiplication.Modular form 570.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
