Show commands:
SageMath
E = EllipticCurve("fk1")
E.isogeny_class()
Elliptic curves in class 369600.fk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 369600.fk1 | 369600fk4 | \([0, -1, 0, -1629469633, -22422540588863]\) | \(233632133015204766393938/29145526885986328125\) | \(59690039062500000000000000000\) | \([2]\) | \(377487360\) | \(4.2512\) | |
| 369600.fk2 | 369600fk2 | \([0, -1, 0, -407017633, 2797866623137]\) | \(7282213870869695463556/912102595400390625\) | \(933993057690000000000000000\) | \([2, 2]\) | \(188743680\) | \(3.9047\) | |
| 369600.fk3 | 369600fk1 | \([0, -1, 0, -393895633, 3009065213137]\) | \(26401417552259125806544/507547744790625\) | \(129932222666400000000000\) | \([2]\) | \(94371840\) | \(3.5581\) | \(\Gamma_0(N)\)-optimal |
| 369600.fk4 | 369600fk3 | \([0, -1, 0, 605482367, 14501354123137]\) | \(11986661998777424518222/51295853620928503125\) | \(-105053908215661574400000000000\) | \([2]\) | \(377487360\) | \(4.2512\) |
Rank
sage: E.rank()
The elliptic curves in class 369600.fk have rank \(0\).
Complex multiplication
The elliptic curves in class 369600.fk do not have complex multiplication.Modular form 369600.2.a.fk
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 LMFDB 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 LMFDB labels.
