Elliptic curve with Cremona label 27a2 (LMFDB label 27.a1)

• Label: 27a2
• Conductor: $27$
• Discriminant: $-177147$
• j-invariant: \( -12288000 \)
• CM: yes (\(D=-27\))
• Rank: $0$
• Torsion structure: trivial

Minimal Weierstrass equation

\(y^2+y=x^3-270x-1708\)  

Mordell-Weil group structure

trivial

Integral points

\(\)  

Invariants

Conductor:	\( 27 \)	=	\(3^{3}\)
Discriminant:	\(-177147 \)	=	\(-1 \cdot 3^{11} \)
j-invariant:	\( -12288000 \)	=	\(-1 \cdot 2^{15} \cdot 3 \cdot 5^{3}\)
Endomorphism ring:	\(\Z\)
Geometric endomorphism ring:	\(\Z[(1+\sqrt{-27})/2]\)	(potential complex multiplication)
Sato-Tate group:	$N(\mathrm{U}(1))$
Faltings height:	\(0.052147932407099199254187350298\dots\)
Stable Faltings height:	\(-0.95491333220533468452478745021\dots\)

BSD invariants

Analytic rank:	\(0\)
Regulator:	\(1\)
Real period:	\(0.58887958342848331910456316655\dots\)
Tamagawa product:	\( 1 \)
Torsion order:	\(1\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form   27.2.a.a

\( q - 2q^{4} - q^{7} + 5q^{13} + 4q^{16} - 7q^{19} + O(q^{20}) \)  

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

Modular degree:	3
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L(E,1) \) ≈ \( 0.58887958342848331910456316654932546833 \)

Local data

This elliptic curve is not semistable. There is only one prime of bad reduction:

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(3\)	\(1\)	\(II^{*}\)	Additive	-1	3	11	0

Galois representations

The mod \( p \) Galois representation has maximal image for all primes \( p < 1000 \) except those listed.

prime	Image of Galois representation
\(3\)	B.1.2

For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=-1\).

$p$-adic data

$p$-adic regulators

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

Iwasawa invariants

$p$	2	3
Reduction type	ss	add
$\lambda$-invariant(s)	0,5	-
$\mu$-invariant(s)	0,0	-

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

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 \(d=\) 3, 9 and 27.
Its isogeny class 27a consists of 4 curves linked by isogenies of degrees dividing 27.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$2$	\(\Q(\sqrt{-3}) \)	\(\Z/3\Z\)	2.0.3.1-81.1-CMa2
$3$	3.1.108.1	\(\Z/2\Z\)	Not in database
$3$	3.1.243.1	\(\Z/3\Z\)	Not in database
$6$	6.0.34992.1	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$6$	6.0.177147.2	\(\Z/3\Z \times \Z/3\Z\)	Not in database
$6$	\(\Q(\zeta_{9})\)	\(\Z/9\Z\)	Not in database
$6$	6.0.177147.1	\(\Z/9\Z\)	Not in database
$9$	9.1.24794911296.1	\(\Z/6\Z\)	Not in database
$12$	12.2.15045919506432.1	\(\Z/4\Z\)	Not in database
$12$	12.0.241162079949.1	\(\Z/21\Z\)	Not in database
$18$	18.0.4052555153018976267.1	\(\Z/3\Z \times \Z/9\Z\)	Not in database
$18$	18.0.2954312706550833698643.2	\(\Z/27\Z\)	Not in database
$18$	\(\Q(\zeta_{27})\)	\(\Z/27\Z\)	Not in database
$18$	18.0.1844362878529525198848.1	\(\Z/6\Z \times \Z/6\Z\)	Not in database
$18$	18.0.2529990231179046912.1	\(\Z/2\Z \times \Z/18\Z\)	Not in database
$18$	18.0.1844362878529525198848.2	\(\Z/2\Z \times \Z/18\Z\)	Not in database

We only show fields where the torsion growth is primitive.