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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 170d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
170.c1 | 170d1 | \([1, 0, 1, -3, 6]\) | \(-1771561/17000\) | \(-17000\) | \([3]\) | \(12\) | \(-0.50590\) | \(\Gamma_0(N)\)-optimal |
170.c2 | 170d2 | \([1, 0, 1, 22, -164]\) | \(1256216039/12577280\) | \(-12577280\) | \([]\) | \(36\) | \(0.043405\) |
Rank
sage: E.rank()
The elliptic curves in class 170d have rank \(0\).
Complex multiplication
The elliptic curves in class 170d do not have complex multiplication.Modular form 170.2.a.d
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.