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SageMath
E = EllipticCurve("dj1")
E.isogeny_class()
Elliptic curves in class 75810.dj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
75810.dj1 | 75810do4 | \([1, 0, 0, -4528386715, 117290267764817]\) | \(218289391029690300712901881/306514992000\) | \(14420267838347952000\) | \([2]\) | \(38707200\) | \(3.8412\) | |
75810.dj2 | 75810do3 | \([1, 0, 0, -296195995, 1652697331025]\) | \(61085713691774408830201/10268551781250000000\) | \(483093065143025531250000000\) | \([2]\) | \(38707200\) | \(3.8412\) | |
75810.dj3 | 75810do2 | \([1, 0, 0, -283026715, 1832608132817]\) | \(53294746224000958661881/1997017344000000\) | \(93951440320760064000000\) | \([2, 2]\) | \(19353600\) | \(3.4946\) | |
75810.dj4 | 75810do1 | \([1, 0, 0, -16868635, 31409942225]\) | \(-11283450590382195961/2530373271552000\) | \(-119043639819016077312000\) | \([4]\) | \(9676800\) | \(3.1480\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 75810.dj have rank \(1\).
Complex multiplication
The elliptic curves in class 75810.dj do not have complex multiplication.Modular form 75810.2.a.dj
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.