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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 10010z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10010.o3 | 10010z1 | \([1, 0, 0, -2190, -35900]\) | \(1161631688686561/121121000000\) | \(121121000000\) | \([6]\) | \(14400\) | \(0.86174\) | \(\Gamma_0(N)\)-optimal |
10010.o4 | 10010z2 | \([1, 0, 0, 2810, -174900]\) | \(2453765252833439/14670296641000\) | \(-14670296641000\) | \([6]\) | \(28800\) | \(1.2083\) | |
10010.o1 | 10010z3 | \([1, 0, 0, -172690, -27636000]\) | \(569541582763202518561/828928100\) | \(828928100\) | \([2]\) | \(43200\) | \(1.4110\) | |
10010.o2 | 10010z4 | \([1, 0, 0, -172640, -27652790]\) | \(-569047017391330383361/687121794969610\) | \(-687121794969610\) | \([2]\) | \(86400\) | \(1.7576\) |
Rank
sage: E.rank()
The elliptic curves in class 10010z have rank \(0\).
Complex multiplication
The elliptic curves in class 10010z do not have complex multiplication.Modular form 10010.2.a.z
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.