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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 170.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
170.a1 | 170b3 | \([1, 0, 1, -4169, -20724]\) | \(8010684753304969/4456448000000\) | \(4456448000000\) | \([2]\) | \(480\) | \(1.1175\) | |
170.a2 | 170b1 | \([1, 0, 1, -2554, 49452]\) | \(1841373668746009/31443200\) | \(31443200\) | \([6]\) | \(160\) | \(0.56817\) | \(\Gamma_0(N)\)-optimal |
170.a3 | 170b2 | \([1, 0, 1, -2474, 52716]\) | \(-1673672305534489/241375690000\) | \(-241375690000\) | \([6]\) | \(320\) | \(0.91474\) | |
170.a4 | 170b4 | \([1, 0, 1, 16311, -159988]\) | \(479958568556831351/289000000000000\) | \(-289000000000000\) | \([2]\) | \(960\) | \(1.4641\) |
Rank
sage: E.rank()
The elliptic curves in class 170.a have rank \(0\).
Complex multiplication
The elliptic curves in class 170.a do not have complex multiplication.Modular form 170.2.a.a
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.