Elliptic curve isogeny class with LMFDB label 75810.dj (Cremona label 75810do)

• Label: 75810.dj
• Number of curves: $4$
• Conductor: $75810$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 75810.dj have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 - T\)
\(3\)	\(1 - T\)
\(5\)	\(1 - T\)
\(7\)	\(1 + T\)
\(19\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(11\)	\( 1 + 4 T + 11 T^{2}\)	1.11.e
\(13\)	\( 1 + 2 T + 13 T^{2}\)	1.13.c
\(17\)	\( 1 - 6 T + 17 T^{2}\)	1.17.ag
\(23\)	\( 1 + 4 T + 23 T^{2}\)	1.23.e
\(29\)	\( 1 + 6 T + 29 T^{2}\)	1.29.g
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 75810.dj do not have complex multiplication.

Modular form 75810.2.a.dj

Isogeny 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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 75810.dj

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