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SageMath
E = EllipticCurve("fe1")
E.isogeny_class()
Elliptic curves in class 35280fe
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 35280.ek7 | 35280fe1 | \([0, 0, 0, -3510507, -2530673894]\) | \(13619385906841/6048000\) | \(2124650495213568000\) | \([2]\) | \(884736\) | \(2.4752\) | \(\Gamma_0(N)\)-optimal |
| 35280.ek6 | 35280fe2 | \([0, 0, 0, -4074987, -1662164966]\) | \(21302308926361/8930250000\) | \(3137179246838784000000\) | \([2, 2]\) | \(1769472\) | \(2.8218\) | |
| 35280.ek5 | 35280fe3 | \([0, 0, 0, -10390107, 9796828426]\) | \(353108405631241/86318776320\) | \(30323616212717792133120\) | \([2]\) | \(2654208\) | \(3.0245\) | |
| 35280.ek8 | 35280fe4 | \([0, 0, 0, 13565013, -12207356966]\) | \(785793873833639/637994920500\) | \(-224126359752656406528000\) | \([2]\) | \(3538944\) | \(3.1684\) | |
| 35280.ek4 | 35280fe5 | \([0, 0, 0, -30746667, 64467598426]\) | \(9150443179640281/184570312500\) | \(64839187476000000000000\) | \([4]\) | \(3538944\) | \(3.1684\) | |
| 35280.ek2 | 35280fe6 | \([0, 0, 0, -154896987, 741955386634]\) | \(1169975873419524361/108425318400\) | \(38089600931258066534400\) | \([2, 2]\) | \(5308416\) | \(3.3711\) | |
| 35280.ek3 | 35280fe7 | \([0, 0, 0, -143607387, 854695590154]\) | \(-932348627918877961/358766164249920\) | \(-126033847311419445085470720\) | \([2]\) | \(10616832\) | \(3.7177\) | |
| 35280.ek1 | 35280fe8 | \([0, 0, 0, -2478296667, 47487362908426]\) | \(4791901410190533590281/41160000\) | \(14459426981314560000\) | \([4]\) | \(10616832\) | \(3.7177\) |
Rank
sage: E.rank()
The elliptic curves in class 35280fe have rank \(1\).
Complex multiplication
The elliptic curves in class 35280fe do not have complex multiplication.Modular form 35280.2.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.
