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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 6699f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6699.f4 | 6699f1 | \([1, 0, 1, 48, 361]\) | \(12600539783/62414583\) | \(-62414583\) | \([2]\) | \(1792\) | \(0.17803\) | \(\Gamma_0(N)\)-optimal |
| 6699.f3 | 6699f2 | \([1, 0, 1, -557, 4475]\) | \(19061979249097/2198953449\) | \(2198953449\) | \([2, 2]\) | \(3584\) | \(0.52460\) | |
| 6699.f2 | 6699f3 | \([1, 0, 1, -2152, -33805]\) | \(1101438820807417/148956693039\) | \(148956693039\) | \([2]\) | \(7168\) | \(0.87117\) | |
| 6699.f1 | 6699f4 | \([1, 0, 1, -8642, 308471]\) | \(71366476613135257/1143673377\) | \(1143673377\) | \([2]\) | \(7168\) | \(0.87117\) |
Rank
sage: E.rank()
The elliptic curves in class 6699f have rank \(0\).
Complex multiplication
The elliptic curves in class 6699f do not have complex multiplication.Modular form 6699.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
