Elliptic curve isogeny class with Cremona label 122304il (LMFDB label 122304.fm)

• Label: 122304il
• Number of curves: $4$
• Conductor: $122304$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 122304il have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1 - T\)
\(7\)	\(1\)
\(13\)	\(1 + T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 - 3 T + 5 T^{2}\)	1.5.ad
\(11\)	\( 1 - T + 11 T^{2}\)	1.11.ab
\(17\)	\( 1 + 7 T + 17 T^{2}\)	1.17.h
\(19\)	\( 1 + T + 19 T^{2}\)	1.19.b
\(23\)	\( 1 - 7 T + 23 T^{2}\)	1.23.ah
\(29\)	\( 1 + 3 T + 29 T^{2}\)	1.29.d
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 122304il do not have complex multiplication.

Modular form 122304.2.a.il

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 Cremona 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 Cremona labels.

Elliptic curves in class 122304il

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
122304.fm3	122304il1	\([0, 1, 0, -7709, 257907]\)	\(420616192/117\)	\(14095291392\)	\([2]\)	\(147456\)	\(0.92921\)	\(\Gamma_0(N)\)-optimal
122304.fm2	122304il2	\([0, 1, 0, -8689, 187151]\)	\(37642192/13689\)	\(26386385485824\)	\([2, 2]\)	\(294912\)	\(1.2758\)
122304.fm4	122304il3	\([0, 1, 0, 26591, 1351391]\)	\(269676572/257049\)	\(-1981910732046336\)	\([2]\)	\(589824\)	\(1.6224\)
122304.fm1	122304il4	\([0, 1, 0, -59649, -5489793]\)	\(3044193988/85293\)	\(657629915185152\)	\([2]\)	\(589824\)	\(1.6224\)