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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 161700.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
161700.p1 | 161700et2 | \([0, -1, 0, -205228, -35721608]\) | \(-25910236240/3993\) | \(-147320642515200\) | \([]\) | \(979776\) | \(1.7305\) | |
161700.p2 | 161700et1 | \([0, -1, 0, 572, -159368]\) | \(560/297\) | \(-10957733740800\) | \([]\) | \(326592\) | \(1.1812\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 161700.p have rank \(0\).
Complex multiplication
The elliptic curves in class 161700.p do not have complex multiplication.Modular form 161700.2.a.p
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.