Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2-18689x-975807\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z-18689xz^2-975807z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-1513836x-715904784\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(-77, 28\right)\)
|
| $\hat{h}(P)$ | ≈ | $2.8247765090957095031793459370$ |
Torsion generators
\( \left(-81, 0\right) \)
Integral points
\( \left(-81, 0\right) \), \((-77,\pm 28)\), \((208,\pm 2023)\)
Invariants
| Conductor: | \( 466752 \) | = | $2^{6} \cdot 3 \cdot 11 \cdot 13 \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
| Discriminant: | $1174094118912 $ | = | $2^{15} \cdot 3 \cdot 11 \cdot 13 \cdot 17^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
| j-invariant: | \( \frac{22032065143304}{35830509} \) | = | $2^{3} \cdot 3^{-1} \cdot 11^{-1} \cdot 13^{-1} \cdot 17^{-4} \cdot 107^{3} \cdot 131^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
|
| 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: | $1.2127576124411230969514705588\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
| Stable Faltings height: | $0.34632363674119146017993040698\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
| $abc$ quality: | $0.8561594679194793\dots$ | |||
| Szpiro ratio: | $3.1501558718941043\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
| Regulator: | $2.8247765090957095031793459370\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
| Real period: | $0.40836030845531858494162060463\dots$ | 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);
|
| Tamagawa product: | $ 4 $ = $ 2\cdot1\cdot1\cdot1\cdot2 $ | 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)
|
| Torsion order: | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
| Analytic order of Ш: | $4$ = $2^2$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
| Special value: | $ L'(E,1) $ ≈ $ 4.6141064262866479083785360880 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 4.614106426 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{4 \cdot 0.408360 \cdot 2.824777 \cdot 4}{2^2} \approx 4.614106426$
Modular invariants
Modular form 466752.2.a.o
For more coefficients, see the Downloads section to the right.
| Modular degree: | 802816 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{5}^{*}$ | Additive | -1 | 6 | 15 | 0 |
| $3$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
| $11$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
| $17$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
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 |
|---|---|---|
| $2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 58344 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 38900 & 1 \\ 19471 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 58337 & 8 \\ 58336 & 9 \end{array}\right),\left(\begin{array}{rr} 21224 & 3 \\ 42437 & 2 \end{array}\right),\left(\begin{array}{rr} 36473 & 36468 \\ 36470 & 7295 \end{array}\right),\left(\begin{array}{rr} 44884 & 1 \\ 53879 & 6 \end{array}\right),\left(\begin{array}{rr} 36464 & 7285 \\ 51047 & 51018 \end{array}\right),\left(\begin{array}{rr} 54913 & 8 \\ 44620 & 33 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 58338 & 58339 \end{array}\right)$.
The torsion field $K:=\Q(E[58344])$ is a degree-$41625591585177600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/58344\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 466752.o
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 233376.p1, its twist by $-8$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.