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SageMath
E = EllipticCurve("fk1")
E.isogeny_class()
Elliptic curves in class 334400fk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 334400.fk1 | 334400fk1 | \([0, 1, 0, -2733, 63413]\) | \(-2258403328/480491\) | \(-480491000000\) | \([]\) | \(373248\) | \(0.96288\) | \(\Gamma_0(N)\)-optimal |
| 334400.fk2 | 334400fk2 | \([0, 1, 0, 19267, -365587]\) | \(790939860992/517504691\) | \(-517504691000000\) | \([]\) | \(1119744\) | \(1.5122\) |
Rank
sage: E.rank()
The elliptic curves in class 334400fk have rank \(1\).
Complex multiplication
The elliptic curves in class 334400fk do not have complex multiplication.Modular form 334400.2.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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
