Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-x^2-3514336x+6233083036\) | (homogenize, simplify) |
\(y^2z=x^3-x^2z-3514336xz^2+6233083036z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-284661243x+4543063549542\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-1082, 93636\right)\) |
$\hat{h}(P)$ | ≈ | $1.3157037069508585875175283515$ |
Torsion generators
\( \left(-2459, 0\right) \)
Integral points
\( \left(-2459, 0\right) \), \((-1082,\pm 93636)\), \((2165,\pm 93636)\), \((8166,\pm 722500)\)
Invariants
Conductor: | \( 450840 \) | = | $2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 17^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-13999606574475600000000 $ | = | $-1 \cdot 2^{10} \cdot 3^{8} \cdot 5^{8} \cdot 13 \cdot 17^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{194204905090564}{566398828125} \) | = | $-1 \cdot 2^{2} \cdot 3^{-8} \cdot 5^{-8} \cdot 13^{-1} \cdot 17^{-1} \cdot 191^{6}$ | 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: | $2.9364073292163219685095482348\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.94217800672159283720375415798\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.084811654986052\dots$ | |||
Szpiro ratio: | $4.503390341528022\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.3157037069508585875175283515\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.11033656659052563574712559182\dots$ | 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\cdot2\cdot2\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)
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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: | $4$ = $2^2$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 4.6454473496123147296096043206 $ | 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
$\displaystyle 4.645447350 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{4 \cdot 0.110337 \cdot 1.315704 \cdot 32}{2^2} \approx 4.645447350$
Modular invariants
Modular form 450840.2.a.g
For more coefficients, see the Downloads section to the right.
Modular degree: | 25952256 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $III^{*}$ | Additive | -1 | 3 | 10 | 0 |
$3$ | $2$ | $I_{8}$ | Non-split multiplicative | 1 | 1 | 8 | 8 |
$5$ | $2$ | $I_{8}$ | Non-split multiplicative | 1 | 1 | 8 | 8 |
$13$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$17$ | $4$ | $I_{1}^{*}$ | Additive | 1 | 2 | 7 | 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 | 8.24.0.94 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 53040 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 53025 & 16 \\ 53024 & 17 \end{array}\right),\left(\begin{array}{rr} 36728 & 1 \\ 24559 & 10 \end{array}\right),\left(\begin{array}{rr} 17681 & 16 \\ 35368 & 129 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 12464 & 53035 \\ 31245 & 14 \end{array}\right),\left(\begin{array}{rr} 13 & 16 \\ 40044 & 40105 \end{array}\right),\left(\begin{array}{rr} 19906 & 13271 \\ 33403 & 26694 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 53036 & 53037 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 52942 & 53027 \end{array}\right),\left(\begin{array}{rr} 21217 & 16 \\ 10616 & 129 \end{array}\right)$.
The torsion field $K:=\Q(E[53040])$ is a degree-$6054631503298560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/53040\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 450840g
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 26520bf3, its twist by $17$.
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.