Elliptic curve isogeny class with LMFDB label 187200.ph (Cremona label 187200gf)

• Label: 187200.ph
• Number of curves: $4$
• Conductor: $187200$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 187200.ph have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1\)
\(5\)	\(1\)
\(13\)	\(1 + T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(7\)	\( 1 - 4 T + 7 T^{2}\)	1.7.ae
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(17\)	\( 1 + 2 T + 17 T^{2}\)	1.17.c
\(19\)	\( 1 - 4 T + 19 T^{2}\)	1.19.ae
\(23\)	\( 1 + 8 T + 23 T^{2}\)	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 187200.ph do not have complex multiplication.

Modular form 187200.2.a.ph

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 187200.ph

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
187200.ph1	187200gf3	\([0, 0, 0, -1246010700, 16928979194000]\)	\(71647584155243142409/10140000\)	\(30277877760000000000\)	\([2]\)	\(47185920\)	\(3.5910\)
187200.ph2	187200gf4	\([0, 0, 0, -89402700, 181083386000]\)	\(26465989780414729/10571870144160\)	\(31567435100539453440000000\)	\([2]\)	\(47185920\)	\(3.5910\)
187200.ph3	187200gf2	\([0, 0, 0, -77882700, 264465146000]\)	\(17496824387403529/6580454400\)	\(19649131551129600000000\)	\([2, 2]\)	\(23592960\)	\(3.2444\)
187200.ph4	187200gf1	\([0, 0, 0, -4154700, 5384954000]\)	\(-2656166199049/2658140160\)	\(-7937163987517440000000\)	\([2]\)	\(11796480\)	\(2.8979\)	\(\Gamma_0(N)\)-optimal