Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-x^2-17177x+872025\) | (homogenize, simplify) |
\(y^2z=x^3-x^2z-17177xz^2+872025z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1391364x+631532160\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-23, 1120\right)\) |
$\hat{h}(P)$ | ≈ | $3.8087510443071059361751846190$ |
Torsion generators
\( \left(75, 0\right) \), \( \left(77, 0\right) \)
Integral points
\( \left(-151, 0\right) \), \((-23,\pm 1120)\), \( \left(75, 0\right) \), \( \left(77, 0\right) \)
Invariants
Conductor: | \( 412224 \) | = | $2^{6} \cdot 3 \cdot 19 \cdot 113$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $169928626176 $ | = | $2^{12} \cdot 3^{2} \cdot 19^{2} \cdot 113^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{136845649240768}{41486481} \) | = | $2^{6} \cdot 3^{-2} \cdot 13^{3} \cdot 19^{-2} \cdot 113^{-2} \cdot 991^{3}$ | 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: | $1.1323626359255407496370384320\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.43921545536559544021980631054\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $3.8087510443071059361751846190\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.99617074321453811944410975553\dots$ | 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);
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Tamagawa product: | $ 32 $ = $ 2^{2}\cdot2\cdot2\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)
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Torsion order: | $4$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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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) $ ≈ $ 7.5883327170531158541915598022 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 7.588332717 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.996171 \cdot 3.808751 \cdot 32}{4^2} \approx 7.588332717$
Modular invariants
Modular form 412224.2.a.bb
For more coefficients, see the Downloads section to the right.
Modular degree: | 565248 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | not computed* (one of 3 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{2}^{*}$ | Additive | -1 | 6 | 12 | 0 |
$3$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$19$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$113$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
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 |
---|---|---|
$2$ | 2Cs | 8.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 51528 = 2^{3} \cdot 3 \cdot 19 \cdot 113 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 12885 & 2 \\ 51520 & 51523 \end{array}\right),\left(\begin{array}{rr} 51525 & 4 \\ 51524 & 5 \end{array}\right),\left(\begin{array}{rr} 25993 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 17177 & 4 \\ 34354 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 25763 & 51524 \\ 51526 & 51519 \end{array}\right),\left(\begin{array}{rr} 40681 & 4 \\ 29834 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[51528])$ is a degree-$30559001703874560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/51528\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 412224bb
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 206112g1, its twist by $-8$.
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.