Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-6x-6\) | (homogenize, simplify) |
\(y^2z=x^3-6xz^2-6z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-6x-6\) | (homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
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)
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j-invariant: | \( -13824 \) | = | $-1 \cdot 2^{9} \cdot 3^{3}$ | 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: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $-0.62917836822511953336480818753\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-1.2504050306721196109222355575\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.2262943855309167\dots$ | |||
Szpiro ratio: | $2.2947427391318267\dots$ |
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: | $1.5188944813290509624956164484\dots$ | 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: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 1.5188944813290509624956164483 $ | 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
$\displaystyle 1.518894481 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.518894 \cdot 1.000000 \cdot 1}{1^2} \approx 1.518894481$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 96 | 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 |
---|---|---|
$3$ | 3Nn | 3.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 24.24.1-12.g.1.1, level \( 24 = 2^{3} \cdot 3 \), index $24$, genus $1$, and generators
$\left(\begin{array}{rr} 7 & 0 \\ 6 & 7 \end{array}\right),\left(\begin{array}{rr} 13 & 12 \\ 12 & 13 \end{array}\right),\left(\begin{array}{rr} 10 & 5 \\ 7 & 10 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 3 \end{array}\right),\left(\begin{array}{rr} 4 & 9 \\ 3 & 7 \end{array}\right),\left(\begin{array}{rr} 11 & 0 \\ 0 & 23 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 18 & 23 \end{array}\right),\left(\begin{array}{rr} 19 & 19 \\ 13 & 14 \end{array}\right)$.
The torsion field $K:=\Q(E[24])$ is a degree-$3072$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24\Z)$.
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 | $6$ | \( 64 = 2^{6} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 1728r consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 864i1, its twist by $24$.
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 |
---|---|---|---|
$3$ | 3.1.108.1 | \(\Z/2\Z\) | not in database |
$6$ | 6.0.34992.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$8$ | 8.2.573308928.1 | \(\Z/3\Z\) | not in database |
$12$ | 12.2.3851755393646592.8 | \(\Z/4\Z\) | not in database |
$16$ | 16.0.328683126924509184.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | add | add | ord | ord | ord | ord | ord | ord | ord | ord | ss | ord | ord | ord | ord |
$\lambda$-invariant(s) | - | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 |
$\mu$-invariant(s) | - | - | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.