Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 5712k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5712.k1 | 5712k1 | \([0, -1, 0, -57, 180]\) | \(1302642688/54621\) | \(873936\) | \([2]\) | \(864\) | \(-0.095161\) | \(\Gamma_0(N)\)-optimal |
| 5712.k2 | 5712k2 | \([0, -1, 0, 28, 588]\) | \(9148592/607257\) | \(-155457792\) | \([2]\) | \(1728\) | \(0.25141\) |
Rank
sage: E.rank()
The elliptic curves in class 5712k have rank \(0\).
Complex multiplication
The elliptic curves in class 5712k do not have complex multiplication.Modular form 5712.2.a.k
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
