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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 6137b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6137.b3 | 6137b1 | \([1, -1, 0, -248, -581]\) | \(35937/17\) | \(799779977\) | \([2]\) | \(1728\) | \(0.40244\) | \(\Gamma_0(N)\)-optimal |
6137.b2 | 6137b2 | \([1, -1, 0, -2053, 35880]\) | \(20346417/289\) | \(13596259609\) | \([2, 2]\) | \(3456\) | \(0.74901\) | |
6137.b1 | 6137b3 | \([1, -1, 0, -32738, 2288159]\) | \(82483294977/17\) | \(799779977\) | \([2]\) | \(6912\) | \(1.0956\) | |
6137.b4 | 6137b4 | \([1, -1, 0, -248, 95445]\) | \(-35937/83521\) | \(-3929319027001\) | \([2]\) | \(6912\) | \(1.0956\) |
Rank
sage: E.rank()
The elliptic curves in class 6137b have rank \(1\).
Complex multiplication
The elliptic curves in class 6137b do not have complex multiplication.Modular form 6137.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.