Elliptic curve isogeny class with Cremona label 3249c (LMFDB label 3249.d)

• Label: 3249c
• Number of curves: $3$
• Conductor: $3249$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 3249c have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(3\)	\(1\)
\(19\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 + T + 2 T^{2}\)	1.2.b
\(5\)	\( 1 + 5 T^{2}\)	1.5.a
\(7\)	\( 1 - T + 7 T^{2}\)	1.7.ab
\(11\)	\( 1 - 2 T + 11 T^{2}\)	1.11.ac
\(13\)	\( 1 - 5 T + 13 T^{2}\)	1.13.af
\(17\)	\( 1 - 4 T + 17 T^{2}\)	1.17.ae
\(23\)	\( 1 - 4 T + 23 T^{2}\)	1.23.ae
\(29\)	\( 1 - 8 T + 29 T^{2}\)	1.29.ai
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 3249c do not have complex multiplication.

Modular form 3249.2.a.c

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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 3249c

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
3249.d3	3249c1	\([0, 0, 1, 2166, -1715]\)	\(32768/19\)	\(-651632497731\)	\([]\)	\(2880\)	\(0.95635\)	\(\Gamma_0(N)\)-optimal
3249.d2	3249c2	\([0, 0, 1, -30324, -2162300]\)	\(-89915392/6859\)	\(-235239331680891\)	\([]\)	\(8640\)	\(1.5057\)
3249.d1	3249c3	\([0, 0, 1, -2499564, -1521053555]\)	\(-50357871050752/19\)	\(-651632497731\)	\([]\)	\(25920\)	\(2.0550\)