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SageMath
E = EllipticCurve("ea1")
E.isogeny_class()
Elliptic curves in class 100800.ea
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
100800.ea1 | 100800cy8 | \([0, 0, 0, -5057748300, 138447122182000]\) | \(4791901410190533590281/41160000\) | \(122903101440000000000\) | \([2]\) | \(42467328\) | \(3.8960\) | |
100800.ea2 | 100800cy6 | \([0, 0, 0, -316116300, 2163135238000]\) | \(1169975873419524361/108425318400\) | \(323756265937305600000000\) | \([2, 2]\) | \(21233664\) | \(3.5494\) | |
100800.ea3 | 100800cy7 | \([0, 0, 0, -293076300, 2491823878000]\) | \(-932348627918877961/358766164249920\) | \(-1071270026191633121280000000\) | \([2]\) | \(42467328\) | \(3.8960\) | |
100800.ea4 | 100800cy5 | \([0, 0, 0, -62748300, 187952182000]\) | \(9150443179640281/184570312500\) | \(551124000000000000000000\) | \([2]\) | \(14155776\) | \(3.3467\) | |
100800.ea5 | 100800cy3 | \([0, 0, 0, -21204300, 28562182000]\) | \(353108405631241/86318776320\) | \(257746484991098880000000\) | \([2]\) | \(10616832\) | \(3.2029\) | |
100800.ea6 | 100800cy2 | \([0, 0, 0, -8316300, -4845962000]\) | \(21302308926361/8930250000\) | \(26665583616000000000000\) | \([2, 2]\) | \(7077888\) | \(3.0001\) | |
100800.ea7 | 100800cy1 | \([0, 0, 0, -7164300, -7378058000]\) | \(13619385906841/6048000\) | \(18059231232000000000\) | \([2]\) | \(3538944\) | \(2.6536\) | \(\Gamma_0(N)\)-optimal |
100800.ea8 | 100800cy4 | \([0, 0, 0, 27683700, -35589962000]\) | \(785793873833639/637994920500\) | \(-1905042624694272000000000\) | \([2]\) | \(14155776\) | \(3.3467\) |
Rank
sage: E.rank()
The elliptic curves in class 100800.ea have rank \(0\).
Complex multiplication
The elliptic curves in class 100800.ea do not have complex multiplication.Modular form 100800.2.a.ea
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.