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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 266910ck
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 266910.ck6 | 266910ck1 | \([1, 0, 0, -8184310140, 326543733507600]\) | \(-60627540058019895893705412918183361/10979320318525440000000000000000\) | \(-10979320318525440000000000000000\) | \([8]\) | \(830472192\) | \(4.6812\) | \(\Gamma_0(N)\)-optimal |
| 266910.ck5 | 266910ck2 | \([1, 0, 0, -136184310140, 19343222133507600]\) | \(279321867933339260687315798724918183361/7185070196198556975129600000000\) | \(7185070196198556975129600000000\) | \([2, 8]\) | \(1660944384\) | \(5.0278\) | |
| 266910.ck2 | 266910ck3 | \([1, 0, 0, -2178935510140, 1237982811464867600]\) | \(1144081181272479239133643526834383042983361/757051842880264964991840000\) | \(757051842880264964991840000\) | \([8]\) | \(3321888768\) | \(5.3743\) | |
| 266910.ck4 | 266910ck4 | \([1, 0, 0, -141433110140, 17771562402147600]\) | \(312879412318855009716463045818793383361/44626411898210828624928814208160000\) | \(44626411898210828624928814208160000\) | \([2, 4]\) | \(3321888768\) | \(5.3743\) | |
| 266910.ck7 | 266910ck5 | \([1, 0, 0, 232359371860, 95786613663828000]\) | \(1387408316478144028202421750997946744639/4766529283100579070606788945040992400\) | \(-4766529283100579070606788945040992400\) | \([4]\) | \(6643777536\) | \(5.7209\) | |
| 266910.ck3 | 266910ck6 | \([1, 0, 0, -599206392140, -160829690671372800]\) | \(23793209304025894275206985511597632711361/2595031464462830007240075021449312400\) | \(2595031464462830007240075021449312400\) | \([2, 2]\) | \(6643777536\) | \(5.7209\) | |
| 266910.ck8 | 266910ck7 | \([1, 0, 0, 797988528760, -799448088117993420]\) | \(56197067952339694092756408383570499642239/308620147391001134944081878878405640180\) | \(-308620147391001134944081878878405640180\) | \([2]\) | \(13287555072\) | \(6.0675\) | |
| 266910.ck1 | 266910ck8 | \([1, 0, 0, -9320773825040, -10952689658836929780]\) | \(89552893167104768935326075943640273911752961/1268853961713740548849854852215284980\) | \(1268853961713740548849854852215284980\) | \([2]\) | \(13287555072\) | \(6.0675\) |
Rank
sage: E.rank()
The elliptic curves in class 266910ck have rank \(1\).
Complex multiplication
The elliptic curves in class 266910ck do not have complex multiplication.Modular form 266910.2.a.ck
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
