Elliptic curve isogeny class with LMFDB label 33.a (Cremona label 33a)

• Label: 33.a
• Number of curves: $4$
• Conductor: $33$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 33.a have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(3\)	\(1 + T\)
\(11\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 - T + 2 T^{2}\)	1.2.ab
\(5\)	\( 1 + 2 T + 5 T^{2}\)	1.5.c
\(7\)	\( 1 - 4 T + 7 T^{2}\)	1.7.ae
\(13\)	\( 1 + 2 T + 13 T^{2}\)	1.13.c
\(17\)	\( 1 + 2 T + 17 T^{2}\)	1.17.c
\(19\)	\( 1 + 19 T^{2}\)	1.19.a
\(23\)	\( 1 - 8 T + 23 T^{2}\)	1.23.ai
\(29\)	\( 1 + 6 T + 29 T^{2}\)	1.29.g
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 33.a do not have complex multiplication.

Modular form 33.2.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.

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

Isogeny graph

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

Elliptic curves in class 33.a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
33.a1	33a3	\([1, 1, 0, -146, 621]\)	\(347873904937/395307\)	\(395307\)	\([4]\)	\(6\)	\(-0.012632\)
33.a2	33a1	\([1, 1, 0, -11, 0]\)	\(169112377/88209\)	\(88209\)	\([2, 2]\)	\(3\)	\(-0.35921\)	\(\Gamma_0(N)\)-optimal
33.a3	33a2	\([1, 1, 0, -6, -9]\)	\(30664297/297\)	\(297\)	\([2]\)	\(6\)	\(-0.70578\)
33.a4	33a4	\([1, 1, 0, 44, 55]\)	\(9090072503/5845851\)	\(-5845851\)	\([2]\)	\(6\)	\(-0.012632\)