Elliptic curve isogeny class with Cremona label 138c (LMFDB label 138.c)

• Label: 138c
• Number of curves: $4$
• Conductor: $138$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 138c have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 - T\)
\(3\)	\(1 + T\)
\(23\)	\(1 + T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 - 2 T + 5 T^{2}\)	1.5.ac
\(7\)	\( 1 + 7 T^{2}\)	1.7.a
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 + 2 T + 13 T^{2}\)	1.13.c
\(17\)	\( 1 - 2 T + 17 T^{2}\)	1.17.ac
\(19\)	\( 1 + 8 T + 19 T^{2}\)	1.19.i
\(29\)	\( 1 + 2 T + 29 T^{2}\)	1.29.c
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 138c do not have complex multiplication.

Modular form 138.2.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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 138c

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
138.c4	138c1	\([1, 1, 1, 3, 3]\)	\(2924207/3312\)	\(-3312\)	\([4]\)	\(8\)	\(-0.63813\)	\(\Gamma_0(N)\)-optimal
138.c3	138c2	\([1, 1, 1, -17, 11]\)	\(545338513/171396\)	\(171396\)	\([2, 2]\)	\(16\)	\(-0.29156\)
138.c2	138c3	\([1, 1, 1, -107, -457]\)	\(135559106353/5037138\)	\(5037138\)	\([2]\)	\(32\)	\(0.055014\)
138.c1	138c4	\([1, 1, 1, -247, 1391]\)	\(1666957239793/301806\)	\(301806\)	\([2]\)	\(32\)	\(0.055014\)