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SageMath
E = EllipticCurve("fk1")
E.isogeny_class()
Elliptic curves in class 323400fk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
323400.fk2 | 323400fk1 | \([0, 1, 0, -6436068, -6266048832]\) | \(7831544736466064/29831377653\) | \(112308215983929504000\) | \([2]\) | \(13713408\) | \(2.7061\) | \(\Gamma_0(N)\)-optimal |
323400.fk1 | 323400fk2 | \([0, 1, 0, -102882768, -401697518832]\) | \(7997484869919944276/116700507\) | \(1757401337349504000\) | \([2]\) | \(27426816\) | \(3.0527\) |
Rank
sage: E.rank()
The elliptic curves in class 323400fk have rank \(1\).
Complex multiplication
The elliptic curves in class 323400fk do not have complex multiplication.Modular form 323400.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.