Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-35358150x+80924422500\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-35358150xz^2+80924422500z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-45824162427x+3775747328647254\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{9}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(3420, 330\right)\) |
$\hat{h}(P)$ | ≈ | $1.0015904901748669486326706209$ |
Torsion generators
\( \left(3450, 600\right) \)
Integral points
\( \left(-6780, 98250\right) \), \( \left(-6780, -91470\right) \), \( \left(-2550, 394350\right) \), \( \left(-2550, -391800\right) \), \( \left(-1200, 349350\right) \), \( \left(-1200, -348150\right) \), \( \left(2400, 98250\right) \), \( \left(2400, -100650\right) \), \( \left(2706, 69792\right) \), \( \left(2706, -72498\right) \), \( \left(3300, 11850\right) \), \( \left(3300, -15150\right) \), \( \left(3420, 330\right) \), \( \left(3420, -3750\right) \), \( \left(3450, 600\right) \), \( \left(3450, -4050\right) \), \( \left(4080, 65670\right) \), \( \left(4080, -69750\right) \), \( \left(4380, 98250\right) \), \( \left(4380, -102630\right) \), \( \left(10050, 855600\right) \), \( \left(10050, -865650\right) \), \( \left(14300, 1573850\right) \), \( \left(14300, -1588150\right) \), \( \left(305700, 168837450\right) \), \( \left(305700, -169143150\right) \)
Invariants
Conductor: | \( 15810 \) | = | $2 \cdot 3 \cdot 5 \cdot 17 \cdot 31$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-169462737117000000000 $ | = | $-1 \cdot 2^{9} \cdot 3^{9} \cdot 5^{9} \cdot 17^{2} \cdot 31^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( -\frac{4888687926204690735691893601}{169462737117000000000} \) | = | $-1 \cdot 2^{-9} \cdot 3^{-9} \cdot 5^{-9} \cdot 17^{-2} \cdot 31^{-3} \cdot 199^{3} \cdot 8528599^{3}$ | 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: | $2.9723210725682762018255037147\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $2.9723210725682762018255037147\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $1.0152510679866893\dots$ | |||
Szpiro ratio: | $6.594348083196218\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $1.0015904901748669486326706209\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.16926160040799491967200472884\dots$ | 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: | $ 4374 $ = $ 3^{2}\cdot3^{2}\cdot3^{2}\cdot2\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: | $9$ | 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) $ ≈ $ 9.1546637033030089191744054887 $ | 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
$\displaystyle 9.154663703 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.169262 \cdot 1.001590 \cdot 4374}{9^2} \approx 9.154663703$
Modular invariants
Modular form 15810.2.a.u
For more coefficients, see the Downloads section to the right.
Modular degree: | 1329696 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $9$ | $I_{9}$ | Split multiplicative | -1 | 1 | 9 | 9 |
$3$ | $9$ | $I_{9}$ | Split multiplicative | -1 | 1 | 9 | 9 |
$5$ | $9$ | $I_{9}$ | Split multiplicative | -1 | 1 | 9 | 9 |
$17$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$31$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
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.1.1 | 9.72.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 11160 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 31 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 5590 & 9 \\ 8361 & 11152 \end{array}\right),\left(\begin{array}{rr} 6850 & 9 \\ 6111 & 11152 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 6687 & 11152 \end{array}\right),\left(\begin{array}{rr} 11143 & 18 \\ 11142 & 19 \end{array}\right),\left(\begin{array}{rr} 5582 & 8379 \\ 2745 & 7858 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 5571 & 11152 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[11160])$ is a degree-$17772576768000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/11160\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 15810u
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{9}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.1.3720.1 | \(\Z/18\Z\) | Not in database |
$6$ | 6.0.51478848000.1 | \(\Z/2\Z \oplus \Z/18\Z\) | Not in database |
$6$ | 6.0.2255067.2 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
$9$ | 9.3.1063615528562424054507000000.2 | \(\Z/27\Z\) | Not in database |
$12$ | deg 12 | \(\Z/36\Z\) | Not in database |
$18$ | 18.0.41687750410085361286002661797888000000.1 | \(\Z/3\Z \oplus \Z/18\Z\) | 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 | split | split | split | ord | ord | ord | nonsplit | ord | ord | ord | split | ord | ord | ord | ord |
$\lambda$-invariant(s) | 6 | 4 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.