# Properties

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

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 0, 1, -30, 63]); // or
magma: E := EllipticCurve("27a4");
sage: E = EllipticCurve([0, 0, 1, -30, 63]) # or
sage: E = EllipticCurve("27a4")
gp: E = ellinit([0, 0, 1, -30, 63]) \\ or
gp: E = ellinit("27a4")

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

## Mordell-Weil group structure

$\Z/{3}\Z$

## Torsion generators

magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp: elltors(E)

$\left(3, 0\right)$

## Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

$\left(3, 0\right)$

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] $N$ = $27$ = $3^{3}$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc $\Delta$ = $-243$ = $-1 \cdot 3^{5}$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j $j$ = $-12288000$ = $-1 \cdot 2^{15} \cdot 3 \cdot 5^{3}$ $\text{End} (E)$ = $\Z[(1+\sqrt{-27})/2]$ (Complex Multiplication) $\text{ST} (E)$ = $N(\mathrm{U}(1))$

## BSD invariants

 magma: Rank(E); sage: E.rank() $r$ = $0$ magma: Regulator(E); sage: E.regulator() $\text{Reg}$ = $1$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] $\Omega$ ≈ $5.29991625086$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] $\prod_p c_p$ = $1$  = $1$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] $\#E_{\text{tor}}$ = $3$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Ш$_{\text{an}}$ = $1$ (exact)

## Modular invariants

### Modular form27.2.1.a

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

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

### Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
9 : curve is not $\Gamma_0(N)$-optimal

### Special L-value attached to the curve

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
sage: r = E.rank();
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp: ar = ellanalyticrank(E);
gp: ar[2]/factorial(ar[1])

$L(E,1)$ ≈ $0.588879583428$

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(E)[5]
prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$3$ $1$ $IV$ Additive -1 3 5 0

## Galois representations

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
sage: rho = E.galois_representation();
sage: [rho.image_type(p) for p in rho.non_surjective()]

The mod $p$ Galois representation has maximal image for all primes $p$ except those listed.

prime Image of Galois representation
$3$ B.1.1

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

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

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

## Iwasawa invariants

$p$ Reduction type 2 3 ss add 0,5 - 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 and 9.
Its isogeny class 27.a consists of 4 curves linked by isogenies of degrees dividing 27.

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

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.3.81.1 $\Z/9\Z$ 3.3.81.1-27.1-a3
3.1.108.1 $\Z/6\Z$ Not in database
6 6.0.34992.1 $\Z/2\Z \times \Z/6\Z$ Not in database
6.0.177147.2 $\Z/3\Z \times \Z/3\Z$ Not in database
6.0.177147.1 $\Z/9\Z$ 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.