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SageMath
E = EllipticCurve("fk1")
E.isogeny_class()
Elliptic curves in class 88200fk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
88200.bh2 | 88200fk1 | \([0, 0, 0, -165375, 57881250]\) | \(-432\) | \(-1157842633500000000\) | \([2]\) | \(1105920\) | \(2.1545\) | \(\Gamma_0(N)\)-optimal |
88200.bh1 | 88200fk2 | \([0, 0, 0, -3472875, 2488893750]\) | \(1000188\) | \(4631370534000000000\) | \([2]\) | \(2211840\) | \(2.5010\) |
Rank
sage: E.rank()
The elliptic curves in class 88200fk have rank \(0\).
Complex multiplication
The elliptic curves in class 88200fk do not have complex multiplication.Modular form 88200.2.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.