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SageMath
E = EllipticCurve("mj1")
E.isogeny_class()
Elliptic curves in class 100800.mj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 100800.mj1 | 100800mw8 | \([0, 0, 0, -5057748300, -138447122182000]\) | \(4791901410190533590281/41160000\) | \(122903101440000000000\) | \([2]\) | \(42467328\) | \(3.8960\) | |
| 100800.mj2 | 100800mw6 | \([0, 0, 0, -316116300, -2163135238000]\) | \(1169975873419524361/108425318400\) | \(323756265937305600000000\) | \([2, 2]\) | \(21233664\) | \(3.5494\) | |
| 100800.mj3 | 100800mw7 | \([0, 0, 0, -293076300, -2491823878000]\) | \(-932348627918877961/358766164249920\) | \(-1071270026191633121280000000\) | \([2]\) | \(42467328\) | \(3.8960\) | |
| 100800.mj4 | 100800mw5 | \([0, 0, 0, -62748300, -187952182000]\) | \(9150443179640281/184570312500\) | \(551124000000000000000000\) | \([2]\) | \(14155776\) | \(3.3467\) | |
| 100800.mj5 | 100800mw3 | \([0, 0, 0, -21204300, -28562182000]\) | \(353108405631241/86318776320\) | \(257746484991098880000000\) | \([2]\) | \(10616832\) | \(3.2029\) | |
| 100800.mj6 | 100800mw2 | \([0, 0, 0, -8316300, 4845962000]\) | \(21302308926361/8930250000\) | \(26665583616000000000000\) | \([2, 2]\) | \(7077888\) | \(3.0001\) | |
| 100800.mj7 | 100800mw1 | \([0, 0, 0, -7164300, 7378058000]\) | \(13619385906841/6048000\) | \(18059231232000000000\) | \([2]\) | \(3538944\) | \(2.6536\) | \(\Gamma_0(N)\)-optimal |
| 100800.mj8 | 100800mw4 | \([0, 0, 0, 27683700, 35589962000]\) | \(785793873833639/637994920500\) | \(-1905042624694272000000000\) | \([2]\) | \(14155776\) | \(3.3467\) |
Rank
sage: E.rank()
The elliptic curves in class 100800.mj have rank \(0\).
Complex multiplication
The elliptic curves in class 100800.mj do not have complex multiplication.Modular form 100800.2.a.mj
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.
