Elliptic curve isogeny class with LMFDB label 122304.gp (Cremona label 122304dz)

• Label: 122304.gp
• Number of curves: $4$
• Conductor: $122304$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 122304.gp have rank \(0\).

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 \(\mathbb{F}_p\)
\(5\)	\( 1 + 5 T^{2}\)	1.5.a
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(17\)	\( 1 - 6 T + 17 T^{2}\)	1.17.ag
\(19\)	\( 1 - 2 T + 19 T^{2}\)	1.19.ac
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 - 6 T + 29 T^{2}\)	1.29.ag
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

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

Modular form 122304.2.a.gp

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 122304.gp

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
122304.gp1	122304dz4	\([0, 1, 0, -146673, 20022351]\)	\(181037698000/14480427\)	\(27911909476319232\)	\([2]\)	\(829440\)	\(1.8990\)
122304.gp2	122304dz3	\([0, 1, 0, -143733, 20926107]\)	\(2725888000000/19773\)	\(2382104245248\)	\([2]\)	\(414720\)	\(1.5524\)
122304.gp3	122304dz2	\([0, 1, 0, -29073, -1912401]\)	\(1409938000/4563\)	\(8795461828608\)	\([2]\)	\(276480\)	\(1.3497\)
122304.gp4	122304dz1	\([0, 1, 0, -2613, -1989]\)	\(16384000/9477\)	\(1141718602752\)	\([2]\)	\(138240\)	\(1.0031\)	\(\Gamma_0(N)\)-optimal