Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-14611518x+20204937364\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-14611518xz^2+20204937364z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-233784291x+1292882207006\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(2675, 14633\right)\)
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| $\hat{h}(P)$ | ≈ | $1.7404893595492376756549154936$ |
Torsion generators
\( \left(2636, -1318\right) \)
Integral points
\( \left(2636, -1318\right) \), \( \left(2675, 14633\right) \), \( \left(2675, -17308\right) \), \( \left(40661, 8143637\right) \), \( \left(40661, -8184298\right) \), \( \left(162380, 65333978\right) \), \( \left(162380, -65496358\right) \)
Invariants
| Conductor: | \( 489762 \) | = | $2 \cdot 3^{2} \cdot 7 \cdot 13^{2} \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $23352779065966253309952 $ | = | $2^{24} \cdot 3^{9} \cdot 7^{2} \cdot 13^{7} \cdot 23 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( \frac{98044243279969897}{6636680773632} \) | = | $2^{-24} \cdot 3^{-3} \cdot 7^{-2} \cdot 13^{-1} \cdot 23^{-1} \cdot 457^{3} \cdot 1009^{3}$ | 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: | $3.0407848067471119232121858490\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $1.2090039836822887094878195098\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $0.9319853202344806\dots$ | |||
| Szpiro ratio: | $4.663950667910543\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $1.7404893595492376756549154936\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.11784127541574745967036828502\dots$ | 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: | $ 64 $ = $ 2\cdot2^{2}\cdot2\cdot2^{2}\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: | $2$ | 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) $ ≈ $ 3.2816237756291139646898162861 $ | 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
$\displaystyle 3.281623776 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.117841 \cdot 1.740489 \cdot 64}{2^2} \approx 3.281623776$
Modular invariants
Modular form 489762.2.a.r
For more coefficients, see the Downloads section to the right.
| Modular degree: | 55738368 | 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)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{24}$ | Non-split multiplicative | 1 | 1 | 24 | 24 |
| $3$ | $4$ | $I_{3}^{*}$ | Additive | -1 | 2 | 9 | 3 |
| $7$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $4$ | $I_{1}^{*}$ | Additive | 1 | 2 | 7 | 1 |
| $23$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7176 = 2^{3} \cdot 3 \cdot 13 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 4489 & 4488 \\ 910 & 4495 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 7169 & 8 \\ 7168 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 7170 & 7171 \end{array}\right),\left(\begin{array}{rr} 940 & 1 \\ 23 & 6 \end{array}\right),\left(\begin{array}{rr} 6283 & 6282 \\ 4498 & 907 \end{array}\right),\left(\begin{array}{rr} 4412 & 7175 \\ 3841 & 7170 \end{array}\right),\left(\begin{array}{rr} 2384 & 7173 \\ 2387 & 7174 \end{array}\right)$.
The torsion field $K:=\Q(E[7176])$ is a degree-$10754978217984$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7176\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 489762r
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 12558c1, its twist by $-39$.
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.