Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3+x^2-3090251153x-66122191120209\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3+x^2z-3090251153xz^2-66122191120209z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-4004965494963x-3084936874422050034\)
|
(homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
| Conductor: | \( 459186 \) | = | $2 \cdot 3 \cdot 7 \cdot 13 \cdot 29^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
| Discriminant: | $-1567619813330128194 $ | = | $-1 \cdot 2 \cdot 3 \cdot 7 \cdot 13^{7} \cdot 29^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
| j-invariant: | \( -\frac{5486773802537974663600129}{2635437714} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-1} \cdot 7^{-1} \cdot 13^{-7} \cdot 41^{3} \cdot 43^{3} \cdot 100043^{3}$ | 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: | $3.7301955890199534387166298799\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
| Stable Faltings height: | $2.0465476740267164251249938637\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
| $abc$ quality: | $1.0593503894068885\dots$ | |||
| Szpiro ratio: | $5.919069638511247\dots$ | |||
BSD invariants
| Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
| Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
| Real period: | $0.010124442418131869381124779936\dots$ | 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: | $ 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)
|
| 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: | $49$ = $7^2$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
| Special value: | $ L(E,1) $ ≈ $ 0.49609767848846159967511421688 $ | 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
$\displaystyle 0.496097678 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{49 \cdot 0.010124 \cdot 1.000000 \cdot 1}{1^2} \approx 0.496097678$
Modular invariants
Modular form 459186.2.a.h
For more coefficients, see the Downloads section to the right.
| Modular degree: | 188776224 | 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)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $1$ | $I_{7}$ | Non-split multiplicative | 1 | 1 | 7 | 7 |
| $29$ | $1$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 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 |
|---|---|---|
| $7$ | 7B.6.3 | 7.24.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 63336 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \cdot 29 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 54057 & 62408 \\ 18067 & 21113 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 19489 & 45878 \\ 1015 & 4467 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 34943 & 0 \\ 0 & 63335 \end{array}\right),\left(\begin{array}{rr} 21113 & 45878 \\ 12383 & 4467 \end{array}\right),\left(\begin{array}{rr} 31669 & 45878 \\ 22939 & 4467 \end{array}\right),\left(\begin{array}{rr} 47503 & 45878 \\ 7105 & 4467 \end{array}\right),\left(\begin{array}{rr} 63323 & 14 \\ 63322 & 15 \end{array}\right)$.
The torsion field $K:=\Q(E[63336])$ is a degree-$27677122961080320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/63336\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 459186.h
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 546.f1, its twist by $29$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.