Elliptic curve isogeny class with Cremona label 3192o (LMFDB label 3192.k)

• Label: 3192o
• Number of curves: $2$
• Conductor: $3192$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 3192o 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 \(\mathbb{F}_p\)
\(5\)	\( 1 + 5 T^{2}\)	1.5.a
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 + 2 T + 13 T^{2}\)	1.13.c
\(17\)	\( 1 + 8 T + 17 T^{2}\)	1.17.i
\(23\)	\( 1 + 4 T + 23 T^{2}\)	1.23.e
\(29\)	\( 1 + 29 T^{2}\)	1.29.a
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

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

Modular form 3192.2.a.o

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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 3192o

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
3192.k1	3192o1	\([0, 1, 0, -35, -66]\)	\(304900096/96957\)	\(1551312\)	\([2]\)	\(768\)	\(-0.10828\)	\(\Gamma_0(N)\)-optimal
3192.k2	3192o2	\([0, 1, 0, 100, -336]\)	\(427694384/477603\)	\(-122266368\)	\([2]\)	\(1536\)	\(0.23829\)