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

• Label: 3192i
• Number of curves: $4$
• Conductor: $3192$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 3192i have rank \(1\).

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 \(\mathbb{F}_p\)
\(5\)	\( 1 - 2 T + 5 T^{2}\)	1.5.ac
\(11\)	\( 1 - 4 T + 11 T^{2}\)	1.11.ae
\(13\)	\( 1 + 6 T + 13 T^{2}\)	1.13.g
\(17\)	\( 1 - 6 T + 17 T^{2}\)	1.17.ag
\(23\)	\( 1 - 8 T + 23 T^{2}\)	1.23.ai
\(29\)	\( 1 + 6 T + 29 T^{2}\)	1.29.g
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

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

Modular form 3192.2.a.i

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 3192i

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
3192.m4	3192i1	\([0, 1, 0, 41, 86]\)	\(464857088/410571\)	\(-6569136\)	\([4]\)	\(512\)	\(-0.0043898\)	\(\Gamma_0(N)\)-optimal
3192.m3	3192i2	\([0, 1, 0, -204, 576]\)	\(3685542352/1432809\)	\(366799104\)	\([2, 2]\)	\(1024\)	\(0.34218\)
3192.m2	3192i3	\([0, 1, 0, -1464, -21600]\)	\(339112345828/8210223\)	\(8407268352\)	\([2]\)	\(2048\)	\(0.68876\)
3192.m1	3192i4	\([0, 1, 0, -2864, 58032]\)	\(2538016415428/872613\)	\(893555712\)	\([2]\)	\(2048\)	\(0.68876\)