Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-215208x+38426976\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-215208xz^2+38426976z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-215208x+38426976\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
| $P$ | = |
\(\left(273, 147\right)\)
|
\(\left(420, 4704\right)\)
|
| $\hat{h}(P)$ | ≈ | $0.98659149759859815358156834978$ | $1.7880026543342290052958891973$ |
Integral points
\((268,\pm 8)\), \((273,\pm 147)\), \((396,\pm 3912)\), \((420,\pm 4704)\), \((128065,\pm 45829259)\)
Invariants
| Conductor: | \( 508032 \) | = | $2^{7} \cdot 3^{4} \cdot 7^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
| Discriminant: | $-1405192126464 $ | = | $-1 \cdot 2^{14} \cdot 3^{6} \cdot 7^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
| j-invariant: | \( -784446336 \) | = | $-1 \cdot 2^{7} \cdot 3^{3} \cdot 61^{3}$ | 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: | $1.6089655115360311475657840078\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
| Stable Faltings height: | $-0.72196741797894987833795245742\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
| $abc$ quality: | $1.0977193254871285\dots$ | |||
| Szpiro ratio: | $3.687820123166492\dots$ | |||
BSD invariants
| Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
| Regulator: | $1.5886949164504111787859943140\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
| Real period: | $0.72251253520417914993236455370\dots$ | 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: | $ 12 $ = $ 2\cdot3\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: | $1$ | 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! $ ≈ $ 13.774223901006937921436650274 $ | 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
$\displaystyle 13.774223901 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.722513 \cdot 1.588695 \cdot 12}{1^2} \approx 13.774223901$
Modular invariants
Modular form 508032.2.a.bj
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1658880 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | Additive | 1 | 7 | 14 | 0 |
| $3$ | $3$ | $IV$ | Additive | 1 | 4 | 6 | 0 |
| $7$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 2G | 4.2.0.1 |
| $5$ | 5S4 | 5.5.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \), index $20$, genus $0$, and generators
$\left(\begin{array}{rr} 104 & 175 \\ 315 & 104 \end{array}\right),\left(\begin{array}{rr} 16 & 1 \\ 303 & 19 \end{array}\right),\left(\begin{array}{rr} 401 & 20 \\ 400 & 21 \end{array}\right),\left(\begin{array}{rr} 419 & 175 \\ 105 & 104 \end{array}\right),\left(\begin{array}{rr} 179 & 0 \\ 0 & 419 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 6 & 85 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 337 & 140 \\ 84 & 1 \end{array}\right),\left(\begin{array}{rr} 281 & 245 \\ 315 & 316 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[420])$ is a degree-$222953472$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/420\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 508032.bj consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 10368.k1, its twist by $-84$.
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.