Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-296366700x+1963777214000\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-296366700xz^2+1963777214000z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-296366700x+1963777214000\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(\frac{258301}{25}, \frac{8533701}{125}\right)\)
|
| $\hat{h}(P)$ | ≈ | $9.3843598711783485052631786116$ |
Torsion generators
\( \left(9940, 0\right) \)
Integral points
\( \left(9940, 0\right) \)
Invariants
| Conductor: | \( 705600 \) | = | $2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
| Discriminant: | $27664719989760000000 $ | = | $2^{16} \cdot 3^{8} \cdot 5^{7} \cdot 7^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
| j-invariant: | \( \frac{32779037733124}{315} \) | = | $2^{2} \cdot 3^{-2} \cdot 5^{-1} \cdot 7^{-1} \cdot 20161^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
|
| 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: | $3.3088897897832137463168966174\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
| Stable Faltings height: | $0.057713373957858314876575131995\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
| $abc$ quality: | $0.9870093428162598\dots$ | |||
| Szpiro ratio: | $5.207986130193352\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
| Regulator: | $9.3843598711783485052631786116\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
| Real period: | $0.14691043126353122881828610638\dots$ | 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);
|
| Tamagawa product: | $ 32 $ = $ 2\cdot2\cdot2^{2}\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)
|
| Torsion order: | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
| Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
| Special value: | $ L'(E,1) $ ≈ $ 11.029282846455900362408501014 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 11.029282846 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.146910 \cdot 9.384360 \cdot 32}{2^2} \approx 11.029282846$
Modular invariants
Modular form 705600.2.a.bww
For more coefficients, see the Downloads section to the right.
| Modular degree: | 75497472 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
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$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}^{*}$ | Additive | 1 | 6 | 16 | 0 |
| $3$ | $2$ | $I_{2}^{*}$ | Additive | -1 | 2 | 8 | 2 |
| $5$ | $4$ | $I_{1}^{*}$ | Additive | 1 | 2 | 7 | 1 |
| $7$ | $2$ | $I_{1}^{*}$ | 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 | 16.24.0.10 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 5 & 4 \\ 1676 & 1677 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1665 & 16 \\ 1664 & 17 \end{array}\right),\left(\begin{array}{rr} 1328 & 1675 \\ 45 & 14 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 1582 & 1667 \end{array}\right),\left(\begin{array}{rr} 547 & 1664 \\ 1376 & 315 \end{array}\right),\left(\begin{array}{rr} 1066 & 1267 \\ 863 & 942 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1432 & 1679 \\ 1121 & 1670 \end{array}\right),\left(\begin{array}{rr} 1667 & 1664 \\ 996 & 935 \end{array}\right)$.
The torsion field $K:=\Q(E[1680])$ is a degree-$5945425920$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1680\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
4, 2, 4, 8 and 8.
Its isogeny class 705600.bww
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 840.j1, its twist by $840$.
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.