Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+y=x^3+6x\) | (homogenize, simplify) |
\(y^2z+yz^2=x^3+6xz^2\) | (dehomogenize, simplify) |
\(y^2=x^3+96x+16\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(2, 4\right)\) |
$\hat{h}(P)$ | ≈ | $0.22596686882569487875122626489$ |
Integral points
\( \left(0, 0\right) \), \( \left(0, -1\right) \), \( \left(2, 4\right) \), \( \left(2, -5\right) \), \( \left(36, 216\right) \), \( \left(36, -217\right) \)
Invariants
Conductor: | \( 171 \) | = | $3^{2} \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-13851 $ | = | $-1 \cdot 3^{6} \cdot 19 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{32768}{19} \) | = | $2^{15} \cdot 19^{-1}$ | 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);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $-0.51586698734273704217752773488\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-1.0651731316767918878751503533\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.22596686882569487875122626489\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $2.3827779033284052041975492616\dots$ | 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: | $ 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'(E,1) $ ≈ $ 1.0768577238443480228145386666 $ | 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
$\displaystyle 1.076857724 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 2.382778 \cdot 0.225967 \cdot 2}{1^2} \approx 1.076857724$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 8 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$19$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
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.1.2 | 27.72.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1026 = 2 \cdot 3^{3} \cdot 19 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 31 & 36 \\ 334 & 421 \end{array}\right),\left(\begin{array}{rr} 994 & 9 \\ 689 & 790 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 973 & 54 \\ 972 & 55 \end{array}\right),\left(\begin{array}{rr} 877 & 345 \\ 755 & 578 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 729 & 703 \end{array}\right)$.
The torsion field $K:=\Q(E[1026])$ is a degree-$179508960$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1026\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 171.b
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 19.a3, its twist by $-3$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | 2.0.3.1-361.2-a5 |
$3$ | 3.1.76.1 | \(\Z/2\Z\) | Not in database |
$3$ | 3.1.1083.1 | \(\Z/3\Z\) | Not in database |
$6$ | 6.0.109744.2 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.0.3518667.2 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$6$ | 6.0.3518667.1 | \(\Z/9\Z\) | Not in database |
$6$ | 6.0.9747.1 | \(\Z/9\Z\) | Not in database |
$6$ | 6.0.155952.1 | \(\Z/6\Z\) | Not in database |
$9$ | 9.1.1544610364992.1 | \(\Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | 12.0.8779890495744.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.43564677551979246963.1 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
$18$ | 18.0.64417171850299425397321728.2 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.64417171850299425397321728.1 | \(\Z/18\Z\) | Not in database |
$18$ | 18.0.494296175215808851968.1 | \(\Z/18\Z\) | Not in database |
$18$ | 18.0.45330602413173669724041216.1 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | ss | add | ord | ord | ord | ord | ord | split | ss | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 2,5 | - | 1 | 1 | 1 | 1 | 1 | 2 | 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 |
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.