Elliptic curve isogeny class with Cremona label 3192a (LMFDB label 3192.i)

• Label: 3192a
• Number of curves: $4$
• Conductor: $3192$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 3192a have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1 + T\)
\(7\)	\(1 + T\)
\(19\)	\(1 - T\)

Good L-factors:

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

See L-function page for more information

Complex multiplication

The elliptic curves in class 3192a do not have complex multiplication.

Modular form 3192.2.a.a

Isogeny matrix

The \((i,j)\)-th 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, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 3192a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
3192.i3	3192a1	\([0, -1, 0, -47, -108]\)	\(733001728/10773\)	\(172368\)	\([2]\)	\(512\)	\(-0.19195\)	\(\Gamma_0(N)\)-optimal
3192.i2	3192a2	\([0, -1, 0, -92, 180]\)	\(340062928/159201\)	\(40755456\)	\([2, 2]\)	\(1024\)	\(0.15462\)
3192.i1	3192a3	\([0, -1, 0, -1232, 17052]\)	\(202119559492/136857\)	\(140141568\)	\([2]\)	\(2048\)	\(0.50119\)
3192.i4	3192a4	\([0, -1, 0, 328, 1020]\)	\(3799448348/2736741\)	\(-2802422784\)	\([2]\)	\(2048\)	\(0.50119\)