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SageMath
E = EllipticCurve("fe1")
E.isogeny_class()
Elliptic curves in class 58800fe
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
58800.cr7 | 58800fe1 | \([0, -1, 0, -9751408, 11719333312]\) | \(13619385906841/6048000\) | \(45538633728000000000\) | \([2]\) | \(2654208\) | \(2.7306\) | \(\Gamma_0(N)\)-optimal |
58800.cr6 | 58800fe2 | \([0, -1, 0, -11319408, 7698981312]\) | \(21302308926361/8930250000\) | \(67240638864000000000000\) | \([2, 2]\) | \(5308416\) | \(3.0772\) | |
58800.cr5 | 58800fe3 | \([0, -1, 0, -28861408, -45346066688]\) | \(353108405631241/86318776320\) | \(649940333777387520000000\) | \([2]\) | \(7962624\) | \(3.2799\) | |
58800.cr8 | 58800fe4 | \([0, -1, 0, 37680592, 56502981312]\) | \(785793873833639/637994920500\) | \(-4803805721721888000000000\) | \([2]\) | \(10616832\) | \(3.4238\) | |
58800.cr4 | 58800fe5 | \([0, -1, 0, -85407408, -298432634688]\) | \(9150443179640281/184570312500\) | \(1389728812500000000000000\) | \([2]\) | \(10616832\) | \(3.4238\) | |
58800.cr2 | 58800fe6 | \([0, -1, 0, -430269408, -3434835218688]\) | \(1169975873419524361/108425318400\) | \(816392338204262400000000\) | \([2, 2]\) | \(15925248\) | \(3.6265\) | |
58800.cr3 | 58800fe7 | \([0, -1, 0, -398909408, -3956791058688]\) | \(-932348627918877961/358766164249920\) | \(-2701342749301685637120000000\) | \([2]\) | \(31850496\) | \(3.9731\) | |
58800.cr1 | 58800fe8 | \([0, -1, 0, -6884157408, -219846607634688]\) | \(4791901410190533590281/41160000\) | \(309915701760000000000\) | \([2]\) | \(31850496\) | \(3.9731\) |
Rank
sage: E.rank()
The elliptic curves in class 58800fe have rank \(1\).
Complex multiplication
The elliptic curves in class 58800fe do not have complex multiplication.Modular form 58800.2.a.fe
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.