Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-769182520x-8312482729868\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-769182520xz^2-8312482729868z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-62303784147x-6059986821426186\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(\frac{4023244}{81}, \frac{6356065870}{729}\right)\)
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| $\hat{h}(P)$ | ≈ | $8.9418486677690845052509076532$ |
Torsion generators
\( \left(32069, 0\right) \)
Integral points
\( \left(32069, 0\right) \)
Invariants
| Conductor: | \( 406560 \) | = | $2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-726753420367734375000000000 $ | = | $-1 \cdot 2^{9} \cdot 3^{7} \cdot 5^{16} \cdot 7^{4} \cdot 11^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( -\frac{55486311952875723077768}{801237030029296875} \) | = | $-1 \cdot 2^{3} \cdot 3^{-7} \cdot 5^{-16} \cdot 7^{-4} \cdot 2999^{3} \cdot 6359^{3}$ | 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: | $3.9589351942870742892038415566\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $2.2401271724679300351099456765\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $1.0644623952717098\dots$ | |||
| Szpiro ratio: | $5.653737040897362\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $8.9418486677690845052509076532\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.014321526823602170336776376328\dots$ | 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: | $ 256 $ = $ 2\cdot1\cdot2^{4}\cdot2\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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Special value: | $ L'(E,1) $ ≈ $ 8.1958992350749616214898214348 $ | 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
$\displaystyle 8.195899235 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.014322 \cdot 8.941849 \cdot 256}{2^2} \approx 8.195899235$
Modular invariants
Modular form 406560.2.a.ch
For more coefficients, see the Downloads section to the right.
| Modular degree: | 275251200 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | Additive | 1 | 5 | 9 | 0 |
| $3$ | $1$ | $I_{7}$ | Non-split multiplicative | 1 | 1 | 7 | 7 |
| $5$ | $16$ | $I_{16}$ | Split multiplicative | -1 | 1 | 16 | 16 |
| $7$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
| $11$ | $4$ | $I_0^{*}$ | 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1585 & 176 \\ 1804 & 705 \end{array}\right),\left(\begin{array}{rr} 529 & 528 \\ 286 & 199 \end{array}\right),\left(\begin{array}{rr} 1841 & 8 \\ 1840 & 9 \end{array}\right),\left(\begin{array}{rr} 1175 & 0 \\ 0 & 1847 \end{array}\right),\left(\begin{array}{rr} 1200 & 1793 \\ 979 & 1134 \end{array}\right),\left(\begin{array}{rr} 452 & 1177 \\ 583 & 342 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1842 & 1843 \end{array}\right)$.
The torsion field $K:=\Q(E[1848])$ is a degree-$40874803200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1848\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 406560ch
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 3360q4, its twist by $-11$.
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.