Elliptic curve isogeny class with LMFDB label 38.a (Cremona label 38a)

• Label: 38.a
• Number of curves: $3$
• Conductor: $38$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 38.a have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 + T\)
\(19\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 - T + 3 T^{2}\)	1.3.ab
\(5\)	\( 1 + 5 T^{2}\)	1.5.a
\(7\)	\( 1 + T + 7 T^{2}\)	1.7.b
\(11\)	\( 1 + 6 T + 11 T^{2}\)	1.11.g
\(13\)	\( 1 - 5 T + 13 T^{2}\)	1.13.af
\(17\)	\( 1 - 3 T + 17 T^{2}\)	1.17.ad
\(23\)	\( 1 - 3 T + 23 T^{2}\)	1.23.ad
\(29\)	\( 1 - 9 T + 29 T^{2}\)	1.29.aj
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 38.a do not have complex multiplication.

Modular form 38.2.a.a

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.

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

Isogeny graph

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

Elliptic curves in class 38.a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
38.a1	38a2	\([1, 0, 1, -86, -2456]\)	\(-69173457625/2550136832\)	\(-2550136832\)	\([]\)	\(18\)	\(0.48517\)
38.a2	38a3	\([1, 0, 1, -16, 22]\)	\(-413493625/152\)	\(-152\)	\([3]\)	\(18\)	\(-0.61344\)
38.a3	38a1	\([1, 0, 1, 9, 90]\)	\(94196375/3511808\)	\(-3511808\)	\([3]\)	\(6\)	\(-0.064137\)	\(\Gamma_0(N)\)-optimal