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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 36414.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
36414.j1 | 36414t6 | \([1, -1, 0, -35682089058, 2594328654970476]\) | \(285531136548675601769470657/17941034271597192\) | \(315695602490628590375454792\) | \([2]\) | \(70778880\) | \(4.5450\) | |
36414.j2 | 36414t4 | \([1, -1, 0, -2234373498, 40374817041780]\) | \(70108386184777836280897/552468975892674624\) | \(9721403100931233571553461824\) | \([2, 2]\) | \(35389440\) | \(4.1985\) | |
36414.j3 | 36414t5 | \([1, -1, 0, -761063058, 92822606071164]\) | \(-2770540998624539614657/209924951154647363208\) | \(-3693899857128042261673746625608\) | \([2]\) | \(70778880\) | \(4.5450\) | |
36414.j4 | 36414t2 | \([1, -1, 0, -235973178, -350583079500]\) | \(82582985847542515777/44772582831427584\) | \(787831253095911235754102784\) | \([2, 2]\) | \(17694720\) | \(3.8519\) | |
36414.j5 | 36414t1 | \([1, -1, 0, -182704698, -949310141004]\) | \(38331145780597164097/55468445663232\) | \(976038733764360595832832\) | \([2]\) | \(8847360\) | \(3.5053\) | \(\Gamma_0(N)\)-optimal |
36414.j6 | 36414t3 | \([1, -1, 0, 910131462, -2758090486284]\) | \(4738217997934888496063/2928751705237796928\) | \(-51535157903033793888281675328\) | \([2]\) | \(35389440\) | \(4.1985\) |
Rank
sage: E.rank()
The elliptic curves in class 36414.j have rank \(0\).
Complex multiplication
The elliptic curves in class 36414.j do not have complex multiplication.Modular form 36414.2.a.j
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.