Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-562542297x+1973424349881\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-562542297xz^2+1973424349881z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-729054816939x+92074273632498726\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{9}\Z\)
Torsion generators
\( \left(22602, 886035\right) \)
Integral points
\( \left(22602, 886035\right) \), \( \left(22602, -908637\right) \), \( \left(31314, 3865539\right) \), \( \left(31314, -3896853\right) \), \( \left(63390, 14835531\right) \), \( \left(63390, -14898921\right) \), \( \left(246936, 122026395\right) \), \( \left(246936, -122273331\right) \)
Invariants
Conductor: | \( 129162 \) | = | $2 \cdot 3 \cdot 11 \cdot 19 \cdot 103$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $9710732423439397326945222144 $ | = | $2^{9} \cdot 3^{18} \cdot 11^{9} \cdot 19 \cdot 103^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{19687440505944790974585155384593}{9710732423439397326945222144} \) | = | $2^{-9} \cdot 3^{-18} \cdot 11^{-9} \cdot 17^{3} \cdot 19^{-1} \cdot 53^{3} \cdot 103^{-3} \cdot 613^{3} \cdot 48889^{3}$ | 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: | $4.0633884135431905280632604916\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $4.0633884135431905280632604916\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0168944199058978\dots$ | |||
Szpiro ratio: | $6.1227478281910725\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.036255854608367262644405807779\dots$ | 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: | $ 4374 $ = $ 3^{2}\cdot( 2 \cdot 3^{2} )\cdot3^{2}\cdot1\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)
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Torsion order: | $9$ | 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: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 1.9578161488518321827979136201 $ | 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
$\displaystyle 1.957816149 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.036256 \cdot 1.000000 \cdot 4374}{9^2} \approx 1.957816149$
Modular invariants
Modular form 129162.2.a.bf
For more coefficients, see the Downloads section to the right.
Modular degree: | 100745856 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $9$ | $I_{9}$ | Split multiplicative | -1 | 1 | 9 | 9 |
$3$ | $18$ | $I_{18}$ | Split multiplicative | -1 | 1 | 18 | 18 |
$11$ | $9$ | $I_{9}$ | Split multiplicative | -1 | 1 | 9 | 9 |
$19$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$103$ | $3$ | $I_{3}$ | 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 \( 1549944 = 2^{3} \cdot 3^{2} \cdot 11 \cdot 19 \cdot 103 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1386793 & 18 \\ 81585 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 387497 & 774990 \\ 774810 & 1248301 \end{array}\right),\left(\begin{array}{rr} 1549927 & 18 \\ 1549926 & 19 \end{array}\right),\left(\begin{array}{rr} 316009 & 18 \\ 1294137 & 163 \end{array}\right),\left(\begin{array}{rr} 774973 & 18 \\ 774981 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 1268137 & 18 \\ 563625 & 163 \end{array}\right),\left(\begin{array}{rr} 387487 & 18 \\ 387495 & 163 \end{array}\right)$.
The torsion field $K:=\Q(E[1549944])$ is a degree-$7511530334077845504000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1549944\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 129162.bf
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 |
---|---|---|---|
$3$ | 3.3.172216.1 | \(\Z/18\Z\) | Not in database |
$6$ | 6.6.5107642516573696.1 | \(\Z/2\Z \oplus \Z/18\Z\) | Not in database |
$6$ | 6.0.3518667.2 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
$9$ | 9.3.253158583950118595989648201666752.1 | \(\Z/27\Z\) | Not in database |
$12$ | deg 12 | \(\Z/36\Z\) | Not in database |
$18$ | 18.0.8720895589440636569891566685704919927873077248.1 | \(\Z/3\Z \oplus \Z/18\Z\) | 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 | 11 | 19 | 103 |
---|---|---|---|---|---|
Reduction type | split | split | split | split | split |
$\lambda$-invariant(s) | 3 | 3 | 1 | 1 | 1 |
$\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.