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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 102.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
102.b1 | 102c3 | \([1, 0, 1, -751, -6046]\) | \(46753267515625/11591221248\) | \(11591221248\) | \([2]\) | \(72\) | \(0.64143\) | |
102.b2 | 102c1 | \([1, 0, 1, -256, 1550]\) | \(1845026709625/793152\) | \(793152\) | \([6]\) | \(24\) | \(0.092120\) | \(\Gamma_0(N)\)-optimal |
102.b3 | 102c2 | \([1, 0, 1, -216, 2062]\) | \(-1107111813625/1228691592\) | \(-1228691592\) | \([6]\) | \(48\) | \(0.43869\) | |
102.b4 | 102c4 | \([1, 0, 1, 1809, -37790]\) | \(655215969476375/1001033261568\) | \(-1001033261568\) | \([2]\) | \(144\) | \(0.98800\) |
Rank
sage: E.rank()
The elliptic curves in class 102.b have rank \(0\).
Complex multiplication
The elliptic curves in class 102.b do not have complex multiplication.Modular form 102.2.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.