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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 6762b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6762.g3 | 6762b1 | \([1, 1, 0, -1740, -29952]\) | \(-4956477625/268272\) | \(-31561932528\) | \([2]\) | \(5760\) | \(0.77309\) | \(\Gamma_0(N)\)-optimal |
6762.g2 | 6762b2 | \([1, 1, 0, -28200, -1834524]\) | \(21081759765625/57132\) | \(6721522668\) | \([2]\) | \(11520\) | \(1.1197\) | |
6762.g4 | 6762b3 | \([1, 1, 0, 9285, -55971]\) | \(752329532375/448524288\) | \(-52768433958912\) | \([2]\) | \(17280\) | \(1.3224\) | |
6762.g1 | 6762b4 | \([1, 1, 0, -37755, -498147]\) | \(50591419971625/28422890688\) | \(3343924666552512\) | \([2]\) | \(34560\) | \(1.6690\) |
Rank
sage: E.rank()
The elliptic curves in class 6762b have rank \(0\).
Complex multiplication
The elliptic curves in class 6762b do not have complex multiplication.Modular form 6762.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.