Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-4x+64\) | (homogenize, simplify) |
\(y^2z=x^3-4xz^2+64z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-4x+64\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(2, 8\right)\) | \(\left(-2, 8\right)\) | \(\left(6, 16\right)\) |
$\hat{h}(P)$ | ≈ | $0.83584539940588044507858488501$ | $0.85591090243313962543875887185$ | $1.1627543299152155216188487321$ |
Integral points
\((-4,\pm 4)\), \((-3,\pm 7)\), \((-2,\pm 8)\), \((0,\pm 8)\), \((2,\pm 8)\), \((5,\pm 13)\), \((6,\pm 16)\), \((10,\pm 32)\), \((13,\pm 47)\), \((16,\pm 64)\), \((28,\pm 148)\), \((42,\pm 272)\), \((62,\pm 488)\), \((66,\pm 536)\), \((128,\pm 1448)\), \((230,\pm 3488)\), \((830,\pm 23912)\), \((1340,\pm 49052)\), \((65408,\pm 16728088)\)
Invariants
Conductor: | \( 27584 \) | = | $2^{6} \cdot 431$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-1765376 $ | = | $-1 \cdot 2^{12} \cdot 431 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{1728}{431} \) | = | $-1 \cdot 2^{6} \cdot 3^{3} \cdot 431^{-1}$ | 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: | $-0.12210831650587831841923483873\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.81525549706582362783646696019\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.7722738443252843\dots$ | |||
Szpiro ratio: | $2.136032328420043\dots$ |
BSD invariants
Analytic rank: | $3$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.62520369698891525671577795317\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $2.1575624937767995019174055251\dots$ | 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: | $ 4 $ = $ 2^{2}\cdot1 $ | 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^{(3)}(E,1)/3! $ ≈ $ 5.3956641903755140602533550919 $ | 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
$\displaystyle 5.395664190 \approx L^{(3)}(E,1)/3! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 2.157562 \cdot 0.625204 \cdot 4}{1^2} \approx 5.395664190$
Modular invariants
Modular form 27584.2.a.a
For more coefficients, see the Downloads section to the right.
Modular degree: | 12032 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{2}^{*}$ | Additive | -1 | 6 | 12 | 0 |
$431$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 862 = 2 \cdot 431 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 2 \\ 7 & 3 \end{array}\right),\left(\begin{array}{rr} 861 & 2 \\ 860 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 861 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[862])$ is a degree-$103280702400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/862\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 27584.a consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 13792.b1, its 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 |
---|---|---|---|
$3$ | 3.1.431.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.80062991.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$8$ | deg 8 | \(\Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | 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 | 431 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | add | ss | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | split |
$\lambda$-invariant(s) | - | 3,5 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 4 |
$\mu$-invariant(s) | - | 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.