Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-1409027x+429934979\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-1409027xz^2+429934979z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-22544427x+27493294246\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{6}\Z\)
Torsion generators
\( \left(987, -494\right) \), \( \left(87, 17506\right) \)
Integral points
\( \left(-1317, 658\right) \), \( \left(-693, 33106\right) \), \( \left(-693, -32414\right) \), \( \left(87, 17506\right) \), \( \left(87, -17594\right) \), \( \left(987, -494\right) \), \( \left(1387, 33106\right) \), \( \left(1387, -34494\right) \), \( \left(4767, 317026\right) \), \( \left(4767, -321794\right) \)
Invariants
Conductor: | \( 8190 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $99313023910464000000 $ | = | $2^{12} \cdot 3^{8} \cdot 5^{6} \cdot 7^{2} \cdot 13^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{424378956393532177129}{136231857216000000} \) | = | $2^{-12} \cdot 3^{-2} \cdot 5^{-6} \cdot 7^{-2} \cdot 13^{-6} \cdot 43^{3} \cdot 174763^{3}$ | 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: | $2.5406594351975826215422343377\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $1.9913532908635277758446117192\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $1.0086784378946845\dots$ | |||
Szpiro ratio: | $6.002754197881709\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.17486258088003773592368203830\dots$ | 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: | $ 3456 $ = $ ( 2^{2} \cdot 3 )\cdot2^{2}\cdot( 2 \cdot 3 )\cdot2\cdot( 2 \cdot 3 ) $ | 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: | $12$ | 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: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
Special value: | $ L(E,1) $ ≈ $ 4.1967019411209056621683689193 $ | 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
$\displaystyle 4.196701941 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.174863 \cdot 1.000000 \cdot 3456}{12^2} \approx 4.196701941$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 331776 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
$3$ | $4$ | $I_{2}^{*}$ | Additive | -1 | 2 | 8 | 2 |
$5$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$7$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$13$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
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 |
---|---|---|
$2$ | 2Cs | 4.12.0.1 |
$3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5460 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 5449 & 12 \\ 5448 & 13 \end{array}\right),\left(\begin{array}{rr} 1813 & 5454 \\ 1826 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 4201 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2737 & 6 \\ 5412 & 5419 \end{array}\right),\left(\begin{array}{rr} 3277 & 12 \\ 3282 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 5444 & 5453 \end{array}\right),\left(\begin{array}{rr} 2341 & 12 \\ 3126 & 73 \end{array}\right)$.
The torsion field $K:=\Q(E[5460])$ is a degree-$304331489280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5460\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 8190.bx
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 2730.o4, its twist by $-3$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{6}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{105}) \) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$4$ | \(\Q(\sqrt{39}, \sqrt{-105})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$4$ | \(\Q(i, \sqrt{39})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$6$ | 6.0.425329947.5 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
$8$ | 8.0.888731494560000.97 | \(\Z/4\Z \oplus \Z/12\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/24\Z\) | Not in database |
$9$ | 9.3.166291899708078963000000.6 | \(\Z/2\Z \oplus \Z/18\Z\) | Not in database |
$12$ | deg 12 | \(\Z/6\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | Not in database |
$18$ | 18.6.174286463337947161193635959002967107543625000000000000.1 | \(\Z/2\Z \oplus \Z/36\Z\) | 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 | 13 |
---|---|---|---|---|---|
Reduction type | split | add | split | split | split |
$\lambda$-invariant(s) | 4 | - | 1 | 1 | 1 |
$\mu$-invariant(s) | 0 | - | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.