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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 570.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
570.i1 | 570i4 | \([1, 1, 1, -492480, 132819117]\) | \(13209596798923694545921/92340\) | \(92340\) | \([2]\) | \(3840\) | \(1.4875\) | |
570.i2 | 570i3 | \([1, 1, 1, -31160, 2011565]\) | \(3345930611358906241/165622259047500\) | \(165622259047500\) | \([2]\) | \(3840\) | \(1.4875\) | |
570.i3 | 570i2 | \([1, 1, 1, -30780, 2065677]\) | \(3225005357698077121/8526675600\) | \(8526675600\) | \([2, 2]\) | \(1920\) | \(1.1409\) | |
570.i4 | 570i1 | \([1, 1, 1, -1900, 32525]\) | \(-758575480593601/40535043840\) | \(-40535043840\) | \([4]\) | \(960\) | \(0.79434\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 570.i have rank \(0\).
Complex multiplication
The elliptic curves in class 570.i do not have complex multiplication.Modular form 570.2.a.i
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.