Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2-13581833x+19258728257\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z-13581833xz^2+19258728257z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-17602055595x+898799256400806\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{10215}{4}, \frac{271139}{8}\right)\) |
$\hat{h}(P)$ | ≈ | $3.1920635407563928675475917384$ |
Torsion generators
\( \left(\frac{8451}{4}, -\frac{8455}{8}\right) \)
Integral points
None
Invariants
Conductor: | \( 443982 \) | = | $2 \cdot 3 \cdot 7 \cdot 11 \cdot 31^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $22513938380275367718 $ | = | $2 \cdot 3^{4} \cdot 7^{6} \cdot 11^{3} \cdot 31^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{312196988566716625}{25367712678} \) | = | $2^{-1} \cdot 3^{-4} \cdot 5^{3} \cdot 7^{-6} \cdot 11^{-3} \cdot 17^{3} \cdot 23^{3} \cdot 347^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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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: | $2.7585503204808277436364723501\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.0415567182382546206718901878\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0145637206421085\dots$ | |||
Szpiro ratio: | $4.682289381304536\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $3.1920635407563928675475917384\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.20433913539087022002619460889\dots$ | 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: | $ 72 $ = $ 1\cdot2\cdot( 2 \cdot 3 )\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: | $2$ | 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);
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Special value: | $ L'(E,1) $ ≈ $ 11.740743072555860570182012254 $ | 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
$\displaystyle 11.740743073 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.204339 \cdot 3.192064 \cdot 72}{2^2} \approx 11.740743073$
Modular invariants
Modular form 443982.2.a.bl
For more coefficients, see the Downloads section to the right.
Modular degree: | 17625600 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | not computed* (one of 4 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$3$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$7$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$11$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$31$ | $2$ | $I_0^{*}$ | 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 |
---|---|---|
$2$ | 2B | 2.3.0.1 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 57288 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \cdot 31 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 16369 & 1860 \\ 13206 & 11161 \end{array}\right),\left(\begin{array}{rr} 44362 & 14787 \\ 10137 & 12928 \end{array}\right),\left(\begin{array}{rr} 19097 & 1860 \\ 47740 & 1 \end{array}\right),\left(\begin{array}{rr} 45354 & 56761 \\ 16709 & 26258 \end{array}\right),\left(\begin{array}{rr} 54778 & 14787 \\ 33573 & 12928 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 57238 & 57279 \end{array}\right),\left(\begin{array}{rr} 48047 & 0 \\ 0 & 57287 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 57277 & 12 \\ 57276 & 13 \end{array}\right)$.
The torsion field $K:=\Q(E[57288])$ is a degree-$18246512148480000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/57288\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 443982.bl
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 462.f1, its twist by $-31$.
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.