Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-x^2-8190x+3146256\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-x^2z-8190xz^2+3146256z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-131043x+201229342\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
| $P$ | = |
\(\left(-128, 1508\right)\)
|
\(\left(-14, 1812\right)\)
|
| $\hat{h}(P)$ | ≈ | $1.5360016717792073788193797177$ | $2.1651318601009241510541095854$ |
Integral points
\( \left(-128, 1508\right) \), \( \left(-128, -1380\right) \), \( \left(-14, 1812\right) \), \( \left(-14, -1798\right) \), \( \left(955, 28944\right) \), \( \left(955, -29899\right) \)
Invariants
| Conductor: | \( 422370 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 13 \cdot 19^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
| Discriminant: | $-4235611235251500 $ | = | $-1 \cdot 2^{2} \cdot 3^{6} \cdot 5^{3} \cdot 13 \cdot 19^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
| j-invariant: | \( -\frac{1771561}{123500} \) | = | $-1 \cdot 2^{-2} \cdot 5^{-3} \cdot 11^{6} \cdot 13^{-1} \cdot 19^{-1}$ | 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: | $1.6782624681116795934950856683\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
| Stable Faltings height: | $-0.34326316580559548220705066611\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
| $abc$ quality: | $0.9179508406339176\dots$ | |||
| Szpiro ratio: | $3.3539067876087114\dots$ | |||
BSD invariants
| Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
| Regulator: | $3.2791941820643038750231516717\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
| Real period: | $0.36137814302222315619446822317\dots$ | 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: | $ 8 $ = $ 2\cdot1\cdot1\cdot1\cdot2^{2} $ | 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^{(2)}(E,1)/2! $ ≈ $ 9.4802328329894086835486170095 $ | 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
$\displaystyle 9.480232833 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.361378 \cdot 3.279194 \cdot 8}{1^2} \approx 9.480232833$
Modular invariants
Modular form 422370.2.a.k
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2592000 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $1$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
| $5$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
| $13$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
| $19$ | $4$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 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 \( 2470 = 2 \cdot 5 \cdot 13 \cdot 19 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 781 & 2 \\ 781 & 3 \end{array}\right),\left(\begin{array}{rr} 951 & 2 \\ 951 & 3 \end{array}\right),\left(\begin{array}{rr} 1977 & 2 \\ 1977 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 2469 & 0 \end{array}\right),\left(\begin{array}{rr} 2469 & 2 \\ 2468 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[2470])$ is a degree-$4646489702400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2470\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 422370k consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 2470c1, its twist by $57$.
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.