Elliptic Curve with LMFDB label 33.a3 (Cremona label 33a2)

• Label: 33.a3
• Conductor: 33
• Discriminant: 297
• j-invariant: \( \frac{30664297}{297} \)
• CM: no
• Rank: 0
• Torsion Structure: \(\Z/{2}\Z\)

Minimal Weierstrass equation

\( y^2 + x y = x^{3} + x^{2} - 6 x - 9 \)

Mordell-Weil group structure

\(\Z/{2}\Z\)

Torsion generators

\( \left(-2, 1\right) \)

Integral points

\( \left(-2, 1\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

Conductor:	\( 33 \)	=	\(3 \cdot 11\)
Discriminant:	\(297 \)	=	\(3^{3} \cdot 11 \)
j-invariant:	\( \frac{30664297}{297} \)	=	\(3^{-3} \cdot 11^{-1} \cdot 313^{3}\)
Endomorphism ring:	\(\Z\)		(no Complex Multiplication)
Sato-Tate Group:	$\mathrm{SU}(2)$

BSD invariants

Rank:	\(0\)
Regulator:	\(1\)
Real period:	\(2.98935659097\)
Tamagawa product:	\( 1 \)  = \( 1\cdot1 \)
Torsion order:	\(2\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 33.2.a.a

\( q + q^{2} - q^{3} - q^{4} - 2q^{5} - q^{6} + 4q^{7} - 3q^{8} + q^{9} - 2q^{10} + q^{11} + q^{12} - 2q^{13} + 4q^{14} + 2q^{15} - q^{16} - 2q^{17} + q^{18} + O(q^{20}) \)

For more coefficients, see the Downloads section to the right.

Modular degree and optimality

6 . This curve is not \( \Gamma_0(N) \)-optimal.

Special L-value

\( L(E,1) \) ≈ \( 0.747339147743 \)

Local data

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(3\)	\(1\)	\( I_{3} \)	Non-split multiplicative	1	1	3	3
\(11\)	\(1\)	\( I_{1} \)	Split multiplicative	-1	1	1	1

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X13d.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 4 & 1 \end{array}\right)$ and has index 12.

The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime	Image of Galois representation
\(2\)	B

$p$-adic data

$p$-adic regulators

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$	2	3	11
Reduction type	ordinary	nonsplit	split
$\lambda$-invariant(s)	0	0	1
$\mu$-invariant(s)	0	0	0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 33.a consists of 4 curves linked by isogenies of degrees dividing 4.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base-change curve
2	\(\Q(\sqrt{33}) \)	\(\Z/2\Z \times \Z/2\Z\)	2.2.33.1-33.1-a4
	\(\Q(\sqrt{-3}) \)	\(\Z/4\Z\)	2.0.3.1-363.1-a3
	\(\Q(\sqrt{-11}) \)	\(\Z/4\Z\)	2.0.11.1-99.2-a3
4	4.0.3993.1	\(\Z/8\Z\)	Not in database
	\(\Q(\sqrt{-3}, \sqrt{-11})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.