Elliptic curve isogeny class with LMFDB label 3136.m (Cremona label 3136t)

• Label: 3136.m
• Number of curves: $4$
• Conductor: $3136$
• CM: \(\Q(\sqrt{-1}) \)
• Rank: $1$

Rank

The elliptic curves in class 3136.m have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(7\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 + 3 T^{2}\)	1.3.a
\(5\)	\( 1 + 2 T + 5 T^{2}\)	1.5.c
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 - 6 T + 13 T^{2}\)	1.13.ag
\(17\)	\( 1 + 2 T + 17 T^{2}\)	1.17.c
\(19\)	\( 1 + 19 T^{2}\)	1.19.a
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 - 10 T + 29 T^{2}\)	1.29.ak
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

Each elliptic curve in class 3136.m has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).

Modular form 3136.2.a.m

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

Elliptic curves in class 3136.m

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality	CM discriminant
3136.m1	3136t3	\([0, 0, 0, -2156, -38416]\)	\(287496\)	\(3855122432\)	\([2]\)	\(1536\)	\(0.70214\)		\(-16\)
3136.m2	3136t4	\([0, 0, 0, -2156, 38416]\)	\(287496\)	\(3855122432\)	\([2]\)	\(1536\)	\(0.70214\)		\(-16\)
3136.m3	3136t2	\([0, 0, 0, -196, 0]\)	\(1728\)	\(481890304\)	\([2, 2]\)	\(768\)	\(0.35557\)		\(-4\)
3136.m4	3136t1	\([0, 0, 0, 49, 0]\)	\(1728\)	\(-7529536\)	\([2]\)	\(384\)	\(0.0089957\)	\(\Gamma_0(N)\)-optimal	\(-4\)