Properties

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

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 1, -30, 63])
 
gp: E = ellinit([0, 0, 1, -30, 63])
 
magma: E := EllipticCurve([0, 0, 1, -30, 63]);
 

\(y^2+y=x^3-30x+63\)  Toggle raw display

Mordell-Weil group structure

$\Z/{3}\Z$

Torsion generators

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

\( \left(3, 0\right) \)  Toggle raw display

Integral points

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

\( \left(3, 0\right) \), \( \left(3, -1\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 27 \)  =  $3^{3}$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-243 $  =  $-1 \cdot 3^{5} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
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: $-0.49715821192695564644343526816\dots$
Stable Faltings height: $-0.95491333220533468452478745021\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $0$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $5.2999162508563498719410684989\dots$
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
Tamagawa product: $ 1 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $3$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $1$ (exact)
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Special value: $ L(E,1) $ ≈ $ 0.58887958342848331910456316654932546833 $

Modular invariants

Modular form   27.2.a.a

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

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 9
$ \Gamma_0(N) $-optimal: no
Manin constant: 3

Local data

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

sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
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

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

The mod-$\ell$ Galois representation has maximal image for all primes $\ell < 1000$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image
$3$ 3B.1.1

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

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
 

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 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 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 \(\Z/2\Z \times \Z/6\Z\) Not in database
$6$ 6.0.177147.2 \(\Z/3\Z \times \Z/3\Z\) 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 \(\Z/3\Z \times \Z/9\Z\) Not in database
$18$ 18.0.2954312706550833698643.2 \(\Z/27\Z\) Not in database
$18$ 18.0.1844362878529525198848.1 \(\Z/6\Z \times \Z/6\Z\) Not in database
$18$ 18.0.1844362878529525198848.2 \(\Z/2\Z \times \Z/18\Z\) Not in database
$18$ 18.0.2529990231179046912.1 \(\Z/2\Z \times \Z/18\Z\) Not in database

We only show fields where the torsion growth is primitive.