Elliptic curve isogeny class with LMFDB label 122304.hz (Cremona label 122304dg)

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

Rank

The elliptic curves in class 122304.hz 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 \(\mathbb{F}_p\)
\(5\)	\( 1 - 2 T + 5 T^{2}\)	1.5.ac
\(11\)	\( 1 - 4 T + 11 T^{2}\)	1.11.ae
\(17\)	\( 1 - 2 T + 17 T^{2}\)	1.17.ac
\(19\)	\( 1 + 19 T^{2}\)	1.19.a
\(23\)	\( 1 + 8 T + 23 T^{2}\)	1.23.i
\(29\)	\( 1 - 2 T + 29 T^{2}\)	1.29.ac
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

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

Modular form 122304.2.a.hz

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 122304.hz

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
122304.hz1	122304dg4	\([0, 1, 0, -84737, 9421887]\)	\(17454600584/93639\)	\(360989809410048\)	\([2]\)	\(589824\)	\(1.6375\)
122304.hz2	122304dg2	\([0, 1, 0, -8297, -41385]\)	\(131096512/74529\)	\(35914802466816\)	\([2, 2]\)	\(294912\)	\(1.2909\)
122304.hz3	122304dg1	\([0, 1, 0, -6092, -184710]\)	\(3321287488/7371\)	\(55500209856\)	\([2]\)	\(147456\)	\(0.94431\)	\(\Gamma_0(N)\)-optimal
122304.hz4	122304dg3	\([0, 1, 0, 32863, -296577]\)	\(1018108216/599781\)	\(-2312229187387392\)	\([2]\)	\(589824\)	\(1.6375\)