Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-261358938x+1635681681972\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-261358938xz^2+1635681681972z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-4181743011x+104679445903198\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(\frac{3318902960949627}{410408078161}, \frac{58265974480731449163600}{262920137520359591}\right)\)
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| $\hat{h}(P)$ | ≈ | $32.541592368972051040535754759$ |
Integral points
None
Invariants
| Conductor: | \( 473382 \) | = | $2 \cdot 3^{2} \cdot 7 \cdot 13 \cdot 17^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-13109754757090885387812864 $ | = | $-1 \cdot 2^{17} \cdot 3^{13} \cdot 7 \cdot 13^{5} \cdot 17^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( -\frac{112205650221491190337}{745029571313664} \) | = | $-1 \cdot 2^{-17} \cdot 3^{-7} \cdot 7^{-1} \cdot 13^{-5} \cdot 4823233^{3}$ | 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);
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| Sato-Tate group: | $\mathrm{SU}(2)$ | |||
| Faltings height: | $3.6551931700318626546271712034\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $1.6892803536696997688047812760\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $1.0312090335959399\dots$ | |||
| Szpiro ratio: | $5.339076629111631\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $32.541592368972051040535754759\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.071249467751728179782019358441\dots$ | 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: | $ 4 $ = $ 1\cdot2^{2}\cdot1\cdot1\cdot1 $ | 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: | $1$ | 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) $ ≈ $ 9.2742845443318318930831165546 $ | 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
$\displaystyle 9.274284544 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.071249 \cdot 32.541592 \cdot 4}{1^2} \approx 9.274284544$
Modular invariants
Modular form 473382.2.a.cq
For more coefficients, see the Downloads section to the right.
| Modular degree: | 167552000 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{17}$ | Non-split multiplicative | 1 | 1 | 17 | 17 |
| $3$ | $4$ | $I_{7}^{*}$ | Additive | -1 | 2 | 13 | 7 |
| $7$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $1$ | $I_{5}$ | Non-split multiplicative | 1 | 1 | 5 | 5 |
| $17$ | $1$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
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 \( 2184 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1457 & 2 \\ 1457 & 3 \end{array}\right),\left(\begin{array}{rr} 2017 & 2 \\ 2017 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1093 & 2 \\ 1093 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2183 & 2 \\ 2182 & 3 \end{array}\right),\left(\begin{array}{rr} 1249 & 2 \\ 1249 & 3 \end{array}\right),\left(\begin{array}{rr} 1639 & 2 \\ 1639 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 2183 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[2184])$ is a degree-$1947721531392$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2184\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 473382cq consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 546e1, its twist by $-51$.
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.