Elliptic curve isogeny class with LMFDB label 34.a (Cremona label 34a)

• Label: 34.a
• Number of curves: $4$
• Conductor: $34$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 34.a have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 - T\)
\(17\)	\(1 + T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 + 2 T + 3 T^{2}\)	1.3.c
\(5\)	\( 1 + 5 T^{2}\)	1.5.a
\(7\)	\( 1 + 4 T + 7 T^{2}\)	1.7.e
\(11\)	\( 1 - 6 T + 11 T^{2}\)	1.11.ag
\(13\)	\( 1 - 2 T + 13 T^{2}\)	1.13.ac
\(19\)	\( 1 + 4 T + 19 T^{2}\)	1.19.e
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 + 29 T^{2}\)	1.29.a
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 34.a do not have complex multiplication.

Modular form 34.2.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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 34.a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
34.a1	34a4	\([1, 0, 0, -113, -329]\)	\(159661140625/48275138\)	\(48275138\)	\([2]\)	\(12\)	\(0.17949\)
34.a2	34a3	\([1, 0, 0, -103, -411]\)	\(120920208625/19652\)	\(19652\)	\([2]\)	\(6\)	\(-0.16709\)
34.a3	34a2	\([1, 0, 0, -43, 105]\)	\(8805624625/2312\)	\(2312\)	\([6]\)	\(4\)	\(-0.36982\)
34.a4	34a1	\([1, 0, 0, -3, 1]\)	\(3048625/1088\)	\(1088\)	\([6]\)	\(2\)	\(-0.71639\)	\(\Gamma_0(N)\)-optimal