Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3-30x+63\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3-30xz^2+63z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-480x+4048\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Torsion generators
\( \left(3, 0\right) \)
Integral points
\( \left(3, 0\right) \), \( \left(3, -1\right) \)
Invariants
| Conductor: | \( 27 \) | = | $3^{3}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-243 $ | = | $-1 \cdot 3^{5} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( -12288000 \) | = | $-1 \cdot 2^{15} \cdot 3 \cdot 5^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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| Endomorphism ring: | $\Z$ | |||
| Geometric endomorphism ring: | \(\Z[(1+\sqrt{-27})/2]\) | (potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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| Sato-Tate group: | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $-0.49715821192695564644343526816\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $-0.95491333220533468452478745021\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $1.23864473399791\dots$ | |||
| Szpiro ratio: | $6.619664953039187\dots$ | |||
BSD invariants
| Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $5.2999162508563498719410684989\dots$ | 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);
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| Tamagawa product: | $ 1 $ | 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)
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| Torsion order: | $3$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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| Analytic order of Ш: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| 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);
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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
For more coefficients, see the Downloads section to the right.
| Modular degree: | 9 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 3 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There is only one prime $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $1$ | $IV$ | 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 $0$.