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SageMath
E = EllipticCurve("fk1")
E.isogeny_class()
Elliptic curves in class 77616.fk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 77616.fk1 | 77616ck4 | \([0, 0, 0, -16772259, 26438400898]\) | \(2970658109581346/2139291\) | \(375764358676912128\) | \([2]\) | \(3145728\) | \(2.6842\) | |
| 77616.fk2 | 77616ck3 | \([0, 0, 0, -2413299, -857050670]\) | \(8849350367426/3314597517\) | \(582205791660690548736\) | \([2]\) | \(3145728\) | \(2.6842\) | |
| 77616.fk3 | 77616ck2 | \([0, 0, 0, -1055019, 407508010]\) | \(1478729816932/38900169\) | \(3416388199807435776\) | \([2, 2]\) | \(1572864\) | \(2.3376\) | |
| 77616.fk4 | 77616ck1 | \([0, 0, 0, 12201, 20534038]\) | \(9148592/8301447\) | \(-182267624416534272\) | \([2]\) | \(786432\) | \(1.9910\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 77616.fk have rank \(0\).
Complex multiplication
The elliptic curves in class 77616.fk do not have complex multiplication.Modular form 77616.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
