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SageMath
E = EllipticCurve("ff1")
E.isogeny_class()
Elliptic curves in class 177870.ff
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
177870.ff1 | 177870dy8 | \([1, 1, 1, -2082457616, -36578228819791]\) | \(4791901410190533590281/41160000\) | \(8578665164463240000\) | \([2]\) | \(79626240\) | \(3.6742\) | |
177870.ff2 | 177870dy6 | \([1, 1, 1, -130156496, -571548803407]\) | \(1169975873419524361/108425318400\) | \(22598262922835645337600\) | \([2, 2]\) | \(39813120\) | \(3.3276\) | |
177870.ff3 | 177870dy7 | \([1, 1, 1, -120670096, -658383514447]\) | \(-932348627918877961/358766164249920\) | \(-74774897848369429827842880\) | \([2]\) | \(79626240\) | \(3.6742\) | |
177870.ff4 | 177870dy5 | \([1, 1, 1, -25835741, -49667231041]\) | \(9150443179640281/184570312500\) | \(38468583825020507812500\) | \([2]\) | \(26542080\) | \(3.1249\) | |
177870.ff5 | 177870dy3 | \([1, 1, 1, -8730576, -7549690191]\) | \(353108405631241/86318776320\) | \(17990764806984412692480\) | \([2]\) | \(19906560\) | \(2.9810\) | |
177870.ff6 | 177870dy2 | \([1, 1, 1, -3424121, 1278863543]\) | \(21302308926361/8930250000\) | \(1861263959789792250000\) | \([2, 2]\) | \(13271040\) | \(2.7783\) | |
177870.ff7 | 177870dy1 | \([1, 1, 1, -2949801, 1948034199]\) | \(13619385906841/6048000\) | \(1260538554778272000\) | \([2]\) | \(6635520\) | \(2.4317\) | \(\Gamma_0(N)\)-optimal |
177870.ff8 | 177870dy4 | \([1, 1, 1, 11398379, 9407522543]\) | \(785793873833639/637994920500\) | \(-132972419815302337924500\) | \([2]\) | \(26542080\) | \(3.1249\) |
Rank
sage: E.rank()
The elliptic curves in class 177870.ff have rank \(0\).
Complex multiplication
The elliptic curves in class 177870.ff do not have complex multiplication.Modular form 177870.2.a.ff
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.