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SageMath
E = EllipticCurve("fn1")
E.isogeny_class()
Elliptic curves in class 176400fn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
176400.lp7 | 176400fn1 | \([0, 0, 0, -87762675, -316334236750]\) | \(13619385906841/6048000\) | \(33197663987712000000000\) | \([2]\) | \(21233664\) | \(3.2799\) | \(\Gamma_0(N)\)-optimal |
176400.lp6 | 176400fn2 | \([0, 0, 0, -101874675, -207770620750]\) | \(21302308926361/8930250000\) | \(49018425731856000000000000\) | \([2, 2]\) | \(42467328\) | \(3.6265\) | |
176400.lp5 | 176400fn3 | \([0, 0, 0, -259752675, 1224603553250]\) | \(353108405631241/86318776320\) | \(473806503323715502080000000\) | \([2]\) | \(63700992\) | \(3.8292\) | |
176400.lp4 | 176400fn4 | \([0, 0, 0, -768666675, 8058449803250]\) | \(9150443179640281/184570312500\) | \(1013112304312500000000000000\) | \([2]\) | \(84934656\) | \(3.9731\) | |
176400.lp8 | 176400fn5 | \([0, 0, 0, 339125325, -1525919620750]\) | \(785793873833639/637994920500\) | \(-3501974371135256352000000000\) | \([2]\) | \(84934656\) | \(3.9731\) | |
176400.lp2 | 176400fn6 | \([0, 0, 0, -3872424675, 92744423329250]\) | \(1169975873419524361/108425318400\) | \(595150014550907289600000000\) | \([2, 2]\) | \(127401984\) | \(4.1758\) | |
176400.lp1 | 176400fn7 | \([0, 0, 0, -61957416675, 5935920363553250]\) | \(4791901410190533590281/41160000\) | \(225928546583040000000000\) | \([2]\) | \(254803968\) | \(4.5224\) | |
176400.lp3 | 176400fn8 | \([0, 0, 0, -3590184675, 106836948769250]\) | \(-932348627918877961/358766164249920\) | \(-1969278864240928829460480000000\) | \([2]\) | \(254803968\) | \(4.5224\) |
Rank
sage: E.rank()
The elliptic curves in class 176400fn have rank \(0\).
Complex multiplication
The elliptic curves in class 176400fn do not have complex multiplication.Modular form 176400.2.a.fn
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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.