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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 90.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
90.b1 | 90b4 | \([1, -1, 1, -218, -269]\) | \(57960603/31250\) | \(615093750\) | \([2]\) | \(48\) | \(0.37752\) | |
90.b2 | 90b2 | \([1, -1, 1, -128, 587]\) | \(8527173507/200\) | \(5400\) | \([6]\) | \(16\) | \(-0.17179\) | |
90.b3 | 90b1 | \([1, -1, 1, -8, 11]\) | \(-1860867/320\) | \(-8640\) | \([6]\) | \(8\) | \(-0.51836\) | \(\Gamma_0(N)\)-optimal |
90.b4 | 90b3 | \([1, -1, 1, 52, -53]\) | \(804357/500\) | \(-9841500\) | \([2]\) | \(24\) | \(0.030944\) |
Rank
sage: E.rank()
The elliptic curves in class 90.b have rank \(0\).
Complex multiplication
The elliptic curves in class 90.b do not have complex multiplication.Modular form 90.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.