Elliptic curve with LMFDB label 99.a2 (Cremona label 99a1)

• Label: 99.a2
• Conductor: $99$
• Discriminant: $297$
• j-invariant: \( \frac{19683}{11} \)
• CM: no
• Rank: $1$
• Torsion structure: \(\Z/{2}\Z\)

Minimal Weierstrass equation

\(y^2+xy+y=x^3-x^2-2x\)

Mordell-Weil group structure

$\Z\times \Z/{2}\Z$

Infinite order Mordell-Weil generator and height

$P$	=	\(\left(0, 0\right)\)
$\hat{h}(P)$	≈	$0.30257138456149499917614303481$

Torsion generators

\( \left(-1, 0\right) \)

Integral points

\( \left(-1, 0\right) \), \( \left(0, 0\right) \), \( \left(0, -1\right) \), \( \left(2, 0\right) \), \( \left(2, -3\right) \), \( \left(3, 2\right) \), \( \left(3, -6\right) \), \( \left(26, 117\right) \), \( \left(26, -144\right) \)

Invariants

Conductor:	\( 99 \)	=	$3^{2} \cdot 11$
Discriminant:	$297 $	=	$3^{3} \cdot 11 $
j-invariant:	\( \frac{19683}{11} \)	=	$3^{9} \cdot 11^{-1}$
Endomorphism ring:	$\Z$
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$
Faltings height:	$-0.83526416397439234511324020037\dots$
Stable Faltings height:	$-1.1099172361414197679620515096\dots$

BSD invariants

Analytic rank:	$1$
Regulator:	$0.30257138456149499917614303481\dots$
Real period:	$4.4984528865882752706337785999\dots$
Tamagawa product:	$ 2 $  = $ 2\cdot1 $
Torsion order:	$2$
Analytic order of Ш:	$1$ (exact)
Special value:	$ L'(E,1) $ ≈ $ 0.68055155913983414332871941401 $

Modular invariants

Modular form   99.2.a.a

\( q - q^{2} - q^{4} - 4 q^{5} - 2 q^{7} + 3 q^{8} + 4 q^{10} - q^{11} - 2 q^{13} + 2 q^{14} - q^{16} + 2 q^{17} - 6 q^{19} + O(q^{20}) \)

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

Modular degree:	4
$ \Gamma_0(N) $-optimal:	yes
Manin constant:	1

Local data

This elliptic curve is not semistable. There are 2 primes of bad reduction:

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord($N$)	ord($\Delta$)	ord$(j)_{-}$
$3$	$2$	$III$	Additive	1	2	3	0
$11$	$1$	$I_{1}$	Non-split multiplicative	1	1	1	1

Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$	mod-$\ell$ image	$\ell$-adic image
$2$	2B	2.3.0.1

$p$-adic regulators

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	ord	add	ord	ord	nonsplit	ord	ord	ord	ord	ord	ord	ord	ord	ord	ord
$\lambda$-invariant(s)	1	-	1	1	1	1	1	1	1	1	1	1	1	1	1
$\mu$-invariant(s)	0	-	0	0	0	0	0	0	0	0	0	0	0	0	0

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 99.a consists of 2 curves linked by isogenies of degree 2.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 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
$2$	\(\Q(\sqrt{33}) \)	\(\Z/2\Z \times \Z/2\Z\)	2.2.33.1-99.1-b2
$4$	4.0.4752.1	\(\Z/4\Z\)	Not in database
$8$	8.4.20663487504.1	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	8.0.2732361984.4	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	8.2.32019867.2	\(\Z/6\Z\)	Not in database
$16$	Deg 16	\(\Z/8\Z\)	Not in database
$16$	16.4.135099050711191781841.2	\(\Z/2\Z \times \Z/6\Z\)	Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.