Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+2\) | (homogenize, simplify) |
\(y^2z=x^3+2z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+2\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-1, 1\right)\) |
$\hat{h}(P)$ | ≈ | $0.75457690318122726442207117846$ |
Integral points
\((-1,\pm 1)\)
Invariants
Conductor: | \( 1728 \) | = | $2^{6} \cdot 3^{3}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-1728 $ | = | $-1 \cdot 2^{6} \cdot 3^{3} $ | 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)
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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: | $-0.69989076598103783743244182590\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-1.3211174284280379149898691959\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $\dots$ | |||
Szpiro ratio: | $2.0\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.75457690318122726442207117846\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $3.7476067207013083074870676651\dots$ | 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: | $1$ | 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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 2.8278574736479477248342302820 $ | 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
$\displaystyle 2.827857474 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 3.747607 \cdot 0.754577 \cdot 1}{1^2} \approx 2.827857474$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 48 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 2 primes $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $II$ | additive | 1 | 6 | 6 | 0 |
$3$ | $1$ | $II$ | additive | 1 | 3 | 3 | 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 | 27.972.55.16 |
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 |
---|---|---|---|
$2$ | additive | $2$ | \( 27 = 3^{3} \) |
$3$ | additive | $2$ | \( 64 = 2^{6} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 1728.n
consists of 4 curves linked by isogenies of
degrees dividing 27.
Twists
The minimal quadratic twist of this elliptic curve is 27.a4, its twist by $8$.
The minimal sextic twist of this elliptic curve is 27.a4, its sextic twist by $8$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{2}) \) | \(\Z/3\Z\) | 2.2.8.1-729.1-d3 |
$2$ | \(\Q(\sqrt{-6}) \) | \(\Z/3\Z\) | not in database |
$3$ | 3.1.108.1 | \(\Z/2\Z\) | not in database |
$4$ | \(\Q(\sqrt{2}, \sqrt{-3})\) | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
$6$ | 6.0.34992.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$6$ | 6.6.3359232.1 | \(\Z/9\Z\) | not in database |
$6$ | 6.2.1492992.4 | \(\Z/6\Z\) | not in database |
$6$ | 6.0.4478976.2 | \(\Z/6\Z\) | not in database |
$12$ | 12.2.962938848411648.4 | \(\Z/4\Z\) | not in database |
$12$ | 12.0.101559956668416.2 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
$12$ | deg 12 | \(\Z/7\Z\) | not in database |
$12$ | 12.0.20061226008576.9 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
$18$ | 18.0.543924745232899335442661376.1 | \(\Z/9\Z\) | not in database |
$18$ | 18.6.1768591357765866863198208.1 | \(\Z/18\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | add | add | ss | ord | ss | ord | ss | ord | ss | ss | ord | ord | ss | ord | ss |
$\lambda$-invariant(s) | - | - | 1,1 | 1 | 1,1 | 1 | 1,1 | 1 | 1,1 | 1,1 | 1 | 1 | 1,1 | 1 | 1,1 |
$\mu$-invariant(s) | - | - | 0,0 | 0 | 0,0 | 0 | 0,0 | 0 | 0,0 | 0,0 | 0 | 0 | 0,0 | 0 | 0,0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.