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SageMath
E = EllipticCurve("fk1")
E.isogeny_class()
Elliptic curves in class 210210.fk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 210210.fk1 | 210210c4 | \([1, 0, 0, -299195, -63015975]\) | \(25176685646263969/57915000\) | \(6813641835000\) | \([2]\) | \(1327104\) | \(1.7060\) | |
| 210210.fk2 | 210210c2 | \([1, 0, 0, -18915, -961983]\) | \(6361447449889/294465600\) | \(34643583374400\) | \([2, 2]\) | \(663552\) | \(1.3594\) | |
| 210210.fk3 | 210210c1 | \([1, 0, 0, -3235, 50945]\) | \(31824875809/8785920\) | \(1033654702080\) | \([2]\) | \(331776\) | \(1.0128\) | \(\Gamma_0(N)\)-optimal |
| 210210.fk4 | 210210c3 | \([1, 0, 0, 10485, -3672663]\) | \(1083523132511/50179392120\) | \(-5903555303525880\) | \([2]\) | \(1327104\) | \(1.7060\) |
Rank
sage: E.rank()
The elliptic curves in class 210210.fk have rank \(1\).
Complex multiplication
The elliptic curves in class 210210.fk do not have complex multiplication.Modular form 210210.2.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.
