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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 163170.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
163170.c1 | 163170ed4 | \([1, -1, 0, -5221890, -4591623240]\) | \(183607808836587409/5330220\) | \(457152293476620\) | \([2]\) | \(5111808\) | \(2.3215\) | |
163170.c2 | 163170ed3 | \([1, -1, 0, -512010, 18862416]\) | \(173078750185489/98393452500\) | \(8438824752722752500\) | \([2]\) | \(5111808\) | \(2.3215\) | |
163170.c3 | 163170ed2 | \([1, -1, 0, -326790, -71487900]\) | \(45000254125009/241491600\) | \(20711797786083600\) | \([2, 2]\) | \(2555904\) | \(1.9749\) | |
163170.c4 | 163170ed1 | \([1, -1, 0, -9270, -2332044]\) | \(-1027243729/26853120\) | \(-2303087939147520\) | \([2]\) | \(1277952\) | \(1.6283\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 163170.c have rank \(2\).
Complex multiplication
The elliptic curves in class 163170.c do not have complex multiplication.Modular form 163170.2.a.c
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.