Elliptic curve isogeny class with LMFDB label 322752.do (Cremona label 322752do)

• Label: 322752.do
• Number of curves: $4$
• Conductor: $322752$
• CM: no
• Rank: $2$

Rank

The elliptic curves in class 322752.do have rank \(2\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1 - T\)
\(41\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 - 2 T + 5 T^{2}\)	1.5.ac
\(7\)	\( 1 + 4 T + 7 T^{2}\)	1.7.e
\(11\)	\( 1 + 4 T + 11 T^{2}\)	1.11.e
\(13\)	\( 1 - 2 T + 13 T^{2}\)	1.13.ac
\(17\)	\( 1 + 2 T + 17 T^{2}\)	1.17.c
\(19\)	\( 1 + 4 T + 19 T^{2}\)	1.19.e
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 + 6 T + 29 T^{2}\)	1.29.g
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 322752.do do not have complex multiplication.

Modular form 322752.2.a.do

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 322752.do

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
322752.do1	322752do4	\([0, 1, 0, -47231617, -124948340257]\)	\(9357915116017/538002\)	\(669926183892807057408\)	\([2]\)	\(30965760\)	\(3.0597\)
322752.do2	322752do2	\([0, 1, 0, -3122177, -1715386785]\)	\(2703045457/544644\)	\(678196877521113317376\)	\([2, 2]\)	\(15482880\)	\(2.7131\)
322752.do3	322752do1	\([0, 1, 0, -970497, 343770975]\)	\(81182737/5904\)	\(7351727669605564416\)	\([2]\)	\(7741440\)	\(2.3665\)	\(\Gamma_0(N)\)-optimal
322752.do4	322752do3	\([0, 1, 0, 6560383, -10234103073]\)	\(25076571983/50863698\)	\(-63336052839610638139392\)	\([2]\)	\(30965760\)	\(3.0597\)