Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+x^2-39798825577x+3055987513005987\) | (homogenize, simplify) |
\(y^2z=x^3+x^2z-39798825577xz^2+3055987513005987z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-3223704871764x+2227824568095979788\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(93422, 12380095\right)\) |
$\hat{h}(P)$ | ≈ | $0.70092992189623474247326923948$ |
Integral points
\((93422,\pm 12380095)\)
Invariants
Conductor: | \( 435344 \) | = | $2^{4} \cdot 7 \cdot 13^{2} \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $705101561230531259038969088 $ | = | $2^{8} \cdot 7^{9} \cdot 13^{9} \cdot 23^{5} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{5642017163771722268092767232}{570626054098424597} \) | = | $2^{10} \cdot 7^{-9} \cdot 13^{-3} \cdot 23^{-5} \cdot 176622007^{3}$ | 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: | $4.5825746763148025189331203789\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $2.8380018772107372779615552438\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $1.0247510582986594\dots$ | |||
Szpiro ratio: | $6.53385857241787\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $0.70092992189623474247326923948\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.039152023124549354752472200165\dots$ | 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: | $ 180 $ = $ 1\cdot3^{2}\cdot2^{2}\cdot5 $ | 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: | $1$ | 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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
Special value: | $ L'(E,1) $ ≈ $ 4.9397084119385920349260716326 $ | 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
$\displaystyle 4.939708412 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.039152 \cdot 0.700930 \cdot 180}{1^2} \approx 4.939708412$
Modular invariants
Modular form 435344.2.a.k
For more coefficients, see the Downloads section to the right.
Modular degree: | 798336000 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_0^{*}$ | Additive | 1 | 4 | 8 | 0 |
$7$ | $9$ | $I_{9}$ | Split multiplicative | -1 | 1 | 9 | 9 |
$13$ | $4$ | $I_{3}^{*}$ | Additive | 1 | 2 | 9 | 3 |
$23$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
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 \( 4186 = 2 \cdot 7 \cdot 13 \cdot 23 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1795 & 2 \\ 1795 & 3 \end{array}\right),\left(\begin{array}{rr} 3823 & 2 \\ 3823 & 3 \end{array}\right),\left(\begin{array}{rr} 1289 & 2 \\ 1289 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 4185 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 4185 & 2 \\ 4184 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[4186])$ is a degree-$42347726733312$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4186\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 435344.k consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 16744.i1, its twist by $-52$.
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.