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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 126.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
126.b1 | 126a6 | \([1, -1, 1, -24575, 1488935]\) | \(2251439055699625/25088\) | \(18289152\) | \([6]\) | \(144\) | \(0.96241\) | |
126.b2 | 126a5 | \([1, -1, 1, -1535, 23591]\) | \(-548347731625/1835008\) | \(-1337720832\) | \([6]\) | \(72\) | \(0.61583\) | |
126.b3 | 126a4 | \([1, -1, 1, -320, 1883]\) | \(4956477625/941192\) | \(686128968\) | \([6]\) | \(48\) | \(0.41310\) | |
126.b4 | 126a2 | \([1, -1, 1, -95, -331]\) | \(128787625/98\) | \(71442\) | \([2]\) | \(16\) | \(-0.13621\) | |
126.b5 | 126a1 | \([1, -1, 1, -5, -7]\) | \(-15625/28\) | \(-20412\) | \([2]\) | \(8\) | \(-0.48278\) | \(\Gamma_0(N)\)-optimal |
126.b6 | 126a3 | \([1, -1, 1, 40, 155]\) | \(9938375/21952\) | \(-16003008\) | \([6]\) | \(24\) | \(0.066527\) |
Rank
sage: E.rank()
The elliptic curves in class 126.b have rank \(0\).
Complex multiplication
The elliptic curves in class 126.b do not have complex multiplication.Modular form 126.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.