Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-748x-40832\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-748xz^2-40832z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-748x-40832\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
| $P$ | = |
\(\left(48, 184\right)\)
|
\(\left(416, 8464\right)\)
|
| $\hat{h}(P)$ | ≈ | $1.2438348222381517538859982612$ | $2.2563834365157533078658734823$ |
Integral points
\((48,\pm 184)\), \((416,\pm 8464)\), \((1632,\pm 65920)\)
Invariants
| Conductor: | \( 445280 \) | = | $2^{5} \cdot 5 \cdot 11^{2} \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
| Discriminant: | $-693468385280 $ | = | $-1 \cdot 2^{12} \cdot 5 \cdot 11^{2} \cdot 23^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
| j-invariant: | \( -\frac{93386304}{1399205} \) | = | $-1 \cdot 2^{6} \cdot 3^{3} \cdot 5^{-1} \cdot 11 \cdot 17^{3} \cdot 23^{-4}$ | 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: | $0.95348961137077334040482293767\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
| Stable Faltings height: | $-0.13930678132223372635606644678\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
| $abc$ quality: | $0.9375105098561078\dots$ | |||
| Szpiro ratio: | $2.6723329848697484\dots$ | |||
BSD invariants
| Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
| Regulator: | $2.7901000604027614167436217921\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
| Real period: | $0.38796271241190227927118752475\dots$ | 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\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! $ ≈ $ 8.6596382986757416464271450703 $ | 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
$\displaystyle 8.659638299 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.387963 \cdot 2.790100 \cdot 8}{1^2} \approx 8.659638299$
Modular invariants
Modular form 445280.2.a.a
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1277952 | 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 $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{3}^{*}$ | additive | -1 | 5 | 12 | 0 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $11$ | $1$ | $II$ | additive | -1 | 2 | 2 | 0 |
| $23$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
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 label 20.2.0.a.1, level \( 20 = 2^{2} \cdot 5 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 19 & 0 \end{array}\right),\left(\begin{array}{rr} 17 & 2 \\ 17 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 11 & 3 \end{array}\right),\left(\begin{array}{rr} 19 & 2 \\ 18 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[20])$ is a degree-$23040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/20\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | additive | $2$ | \( 605 = 5 \cdot 11^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 89056 = 2^{5} \cdot 11^{2} \cdot 23 \) |
| $11$ | additive | $32$ | \( 3680 = 2^{5} \cdot 5 \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 19360 = 2^{5} \cdot 5 \cdot 11^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 445280.a consists of this curve only.
Twists
This elliptic curve is its own minimal quadratic twist.
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.