Elliptic curve with LMFDB label 27.a2 (Cremona label 27a4)

• Label: 27.a2
• Conductor: $27$
• Discriminant: $-243$
• j-invariant: $-12288000$
• CM: yes ($D=-27$)
• Rank: $0$
• Torsion structure: $\Z/{3}\Z$

Minimal Weierstrass equation

$y^2+y=x^3-30x+63$

(, )

Mordell-Weil group structure

$\Z/{3}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(3, 0)$	$0$	$3$

Integral points

$\left(3, 0\right)$, $\left(3, -1\right)$

Invariants

Conductor:	$N$	=	$27$	=	$3^{3}$
Discriminant:	$\Delta$	=	$-243$	=	$-1 \cdot 3^{5}$
j-invariant:	$j$	=	$-12288000$	=	$-1 \cdot 2^{15} \cdot 3 \cdot 5^{3}$
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z[(1+\sqrt{-27})/2]$    (potential complex multiplication)
Sato-Tate group:	$\mathrm{ST}(E)$	=	$N(\mathrm{U}(1))$
Faltings height:	$h_{\mathrm{Faltings}}$	≈	$-0.49715821192695564644343526816$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-0.95491333220533468452478745021$
$abc$ quality:	$Q$	≈	$1.23864473399791$
Szpiro ratio:	$\sigma_{m}$	≈	$6.619664953039187$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$
Real period:	$\Omega$	≈	$5.2999162508563498719410684989$
Tamagawa product:	$\prod_{p}c_p$	=	$1$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$3$
Special value:	$L(E,1)$	≈	$0.58887958342848331910456316655$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$    (exact)

BSD formula

$\displaystyle 0.588879583 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 5.299916 \cdot 1.000000 \cdot 1}{3^2} \approx 0.588879583$

Modular invariants

Modular form   27.2.a.a

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

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

Modular degree:	9
$\Gamma_0(N)$-optimal:	no
Manin constant:	3

Local data at primes of bad reduction

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

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$3$	$1$	$IV$	additive	-1	3	5	0

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
$3$	3B.1.1	27.648.13.25

The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.

$\ell$	Reduction type	Serre weight	Serre conductor
$3$	additive	$8$	$1$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3, 9 and 27.
Its isogeny class 27.a consists of 4 curves linked by isogenies of degrees dividing 27.

Twists

This elliptic curve is its own minimal quadratic twist.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$3$	3.1.108.1	$\Z/6\Z$	not in database
$3$	$\Q(\zeta_{9})^+$	$\Z/9\Z$	3.3.81.1-27.1-a3
$6$	6.0.34992.1	$\Z/2\Z \oplus \Z/6\Z$	not in database
$6$	6.0.177147.2	$\Z/3\Z \oplus \Z/3\Z$	not in database
$6$	6.0.177147.1	$\Z/9\Z$	not in database
$9$	$\Q(\zeta_{27})^+$	$\Z/27\Z$	not in database
$9$	9.3.918330048.1	$\Z/18\Z$	not in database
$12$	12.2.15045919506432.1	$\Z/12\Z$	not in database
$12$	12.0.241162079949.1	$\Z/21\Z$	not in database
$18$	18.0.4052555153018976267.1	$\Z/3\Z \oplus \Z/9\Z$	not in database
$18$	18.0.2954312706550833698643.2	$\Z/27\Z$	not in database
$18$	18.0.1844362878529525198848.1	$\Z/6\Z \oplus \Z/6\Z$	not in database
$18$	18.0.1844362878529525198848.2	$\Z/2\Z \oplus \Z/18\Z$	not in database
$18$	18.0.2529990231179046912.1	$\Z/2\Z \oplus \Z/18\Z$	not in database

We only show fields where the torsion growth is primitive.

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.

$p$-adic regulators

All $p$-adic regulators are identically $1$ since the rank is $0$.