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

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

Rank

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

Modular form 33.2.a.a

Isogeny matrix

The \(i,j\) 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.

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

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 33a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
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.a1	33a3	\([1, 1, 0, -146, 621]\)	\(347873904937/395307\)	\(395307\)	\([4]\)	\(6\)	\(-0.012632\)
33.a4	33a4	\([1, 1, 0, 44, 55]\)	\(9090072503/5845851\)	\(-5845851\)	\([2]\)	\(6\)	\(-0.012632\)