This is a model for the Fermat cubic curve $X^3+Y^3=Z^3$ and for the modular curve $X_0(27)$.
Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+y=x^3-7\)
|
(homogenize, simplify) |
|
\(y^2z+yz^2=x^3-7z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-432\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Torsion generators
\( \left(3, 4\right) \)
Integral points
\( \left(3, 4\right) \), \( \left(3, -5\right) \)
Invariants
| Conductor: | \( 27 \) | = | $3^{3}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
| Discriminant: | $-19683 $ | = | $-1 \cdot 3^{9} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
| j-invariant: | \( 0 \) | = | $0$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
|
| Endomorphism ring: | $\Z$ | |||
| Geometric endomorphism ring: | \(\Z[(1+\sqrt{-3})/2]\) | (potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
| |
| Sato-Tate group: | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $-0.49715821192695564644343526816\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
| Stable Faltings height: | $-1.3211174284280379149898691959\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
BSD invariants
| Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
| Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
| Real period: | $1.7666387502854499573136894996\dots$ | comment: Real Period
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
|
| Tamagawa product: | $ 3 $ = $ 3 $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
|
| Torsion order: | $3$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
| Analytic order of Ш: | $1$ (exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
| Special value: | $ L(E,1) $ ≈ $ 0.58887958342848331910456316655 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 0.588879583 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.766639 \cdot 1.000000 \cdot 3}{3^2} \approx 0.588879583$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There is only one prime of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $3$ | $3$ | $IV^{*}$ | Additive | -1 | 3 | 9 | 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 |
|---|---|---|
| $3$ | 3Cs.1.1 | 27.1944.55.37 |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 27a
consists of 4 curves linked by isogenies of
degrees dividing 27.
Twists
The minimal quadratic twist of this elliptic curve is 27a3, its twist by $-3$.
The minimal sextic twist of this elliptic curve is 27.a4, its sextic twist by $-27$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z \oplus \Z/3\Z\) | 2.0.3.1-81.1-CMa1 |
| $3$ | 3.1.108.1 | \(\Z/6\Z\) | Not in database |
| $6$ | 6.0.34992.1 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
| $6$ | \(\Q(\zeta_{9})\) | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
| $9$ | 9.3.1162261467.1 | \(\Z/9\Z\) | Not in database |
| $12$ | 12.2.15045919506432.1 | \(\Z/12\Z\) | Not in database |
| $12$ | 12.0.241162079949.1 | \(\Z/3\Z \oplus \Z/21\Z\) | Not in database |
| $18$ | 18.0.4052555153018976267.1 | \(\Z/9\Z \oplus \Z/9\Z\) | Not in database |
| $18$ | 18.0.2529990231179046912.1 | \(\Z/6\Z \oplus \Z/18\Z\) | 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 $0$.