Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-531332x+148316176\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-531332xz^2+148316176z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-688606947x+6930168608286\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(272, 4764\right)\) | \(\left(1497, 51314\right)\) |
$\hat{h}(P)$ | ≈ | $0.34825732455688261905216547925$ | $0.53073074290148979814194652763$ |
Integral points
\( \left(-288, 16804\right) \), \( \left(-288, -16516\right) \), \( \left(-120, 14564\right) \), \( \left(-120, -14444\right) \), \( \left(-48, 13204\right) \), \( \left(-48, -13156\right) \), \( \left(272, 4764\right) \), \( \left(272, -5036\right) \), \( \left(392, 484\right) \), \( \left(392, -876\right) \), \( \left(447, 389\right) \), \( \left(447, -836\right) \), \( \left(545, 4309\right) \), \( \left(545, -4854\right) \), \( \left(647, 8389\right) \), \( \left(647, -9036\right) \), \( \left(1497, 51314\right) \), \( \left(1497, -52811\right) \), \( \left(2297, 103989\right) \), \( \left(2297, -106286\right) \), \( \left(2967, 155684\right) \), \( \left(2967, -158651\right) \), \( \left(34817, 6477909\right) \), \( \left(34817, -6512726\right) \), \( \left(148007, 56866324\right) \), \( \left(148007, -57014331\right) \)
Invariants
Conductor: | \( 141610 \) | = | $2 \cdot 5 \cdot 7^{2} \cdot 17^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $68783134903000000 $ | = | $2^{6} \cdot 5^{6} \cdot 7^{7} \cdot 17^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{1688258640889}{7000000} \) | = | $2^{-6} \cdot 5^{-6} \cdot 7^{-1} \cdot 17^{2} \cdot 1801^{3}$ | 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: | $2.0859449962744698279893934144\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $0.16858547372807448202020550339\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $0.9463713009363851\dots$ | |||
Szpiro ratio: | $4.313612455895603\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $0.15810590808512237794365197834\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.34879070768138249527130038806\dots$ | 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: | $ 144 $ = $ 2\cdot( 2 \cdot 3 )\cdot2^{2}\cdot3 $ | 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: | $1$ | 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^{(2)}(E,1)/2! $ ≈ $ 7.9410055060249125915834910324 $ | 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
$\displaystyle 7.941005506 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.348791 \cdot 0.158106 \cdot 144}{1^2} \approx 7.941005506$
Modular invariants
Modular form 141610.2.a.o
For more coefficients, see the Downloads section to the right.
Modular degree: | 1492992 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
$5$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$7$ | $4$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
$17$ | $3$ | $IV$ | Additive | -1 | 2 | 4 | 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 |
---|---|---|
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 84 = 2^{2} \cdot 3 \cdot 7 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 38 & 51 \\ 31 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 79 & 6 \\ 78 & 7 \end{array}\right),\left(\begin{array}{rr} 11 & 78 \\ 33 & 65 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 43 & 6 \\ 45 & 19 \end{array}\right)$.
The torsion field $K:=\Q(E[84])$ is a degree-$580608$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/84\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 141610bt
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 20230f2, its twist by $-7$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{21}) \) | \(\Z/3\Z\) | Not in database |
$3$ | 3.3.8092.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.6.1833452992.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.0.1023324598863.3 | \(\Z/3\Z\) | Not in database |
$6$ | 6.6.12375807696.1 | \(\Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.4389349772362143513053618986935589466112.2 | \(\Z/6\Z\) | Not in database |
$18$ | 18.6.430462629391562055402156322344612999549000000000000.1 | \(\Z/9\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | nonsplit | ord | split | add | ss | ord | add | ord | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 5 | 4 | 3 | - | 2,2 | 2 | - | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
$\mu$-invariant(s) | 0 | 0 | 0 | - | 0,0 | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.