Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-210177x+37101321\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-210177xz^2+37101321z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-272389419x+1731816400806\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{9}\Z\)
Torsion generators
\( \left(270, 99\right) \)
Integral points
\( \left(-198, 8523\right) \), \( \left(-198, -8325\right) \), \( \left(234, 747\right) \), \( \left(234, -981\right) \), \( \left(270, 99\right) \), \( \left(270, -369\right) \), \( \left(426, 4779\right) \), \( \left(426, -5205\right) \)
Invariants
Conductor: | \( 6942 \) | = | $2 \cdot 3 \cdot 13 \cdot 89$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-1008907331567616 $ | = | $-1 \cdot 2^{18} \cdot 3^{9} \cdot 13^{3} \cdot 89 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{1026784744653907139473}{1008907331567616} \) | = | $-1 \cdot 2^{-18} \cdot 3^{-9} \cdot 13^{-3} \cdot 17^{3} \cdot 53^{3} \cdot 89^{-1} \cdot 11197^{3}$ | 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: | $1.7992893533677160885260591299\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.7992893533677160885260591299\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9841133281365513\dots$ | |||
Szpiro ratio: | $5.4698165718887575\dots$ |
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: | $0.49089595966514646730968877636\dots$ | 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: | $ 486 $ = $ ( 2 \cdot 3^{2} )\cdot3^{2}\cdot3\cdot1 $ | 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) $ ≈ $ 2.9453757579908788038581326582 $ | 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
$\displaystyle 2.945375758 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.490896 \cdot 1.000000 \cdot 486}{9^2} \approx 2.945375758$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 65664 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $18$ | $I_{18}$ | Split multiplicative | -1 | 1 | 18 | 18 |
$3$ | $9$ | $I_{9}$ | Split multiplicative | -1 | 1 | 9 | 9 |
$13$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$89$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 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 |
---|---|---|
$3$ | 3B.1.1 | 9.72.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 20826 = 2 \cdot 3^{2} \cdot 13 \cdot 89 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 20809 & 18 \\ 20808 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 19234 & 9 \\ 16011 & 20818 \end{array}\right),\left(\begin{array}{rr} 19432 & 9 \\ 9585 & 20818 \end{array}\right),\left(\begin{array}{rr} 2 & 9 \\ 20781 & 11368 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[20826])$ is a degree-$263358145290240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/20826\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 6942.l
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 |
---|---|---|---|
$3$ | 3.1.3471.1 | \(\Z/18\Z\) | Not in database |
$6$ | 6.0.41818056111.1 | \(\Z/2\Z \oplus \Z/18\Z\) | Not in database |
$6$ | 6.0.1694040507.1 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
$9$ | 9.3.278111492153792451640512.2 | \(\Z/27\Z\) | Not in database |
$12$ | deg 12 | \(\Z/36\Z\) | Not in database |
$18$ | 18.0.185870941736208504484254034012694964027.1 | \(\Z/3\Z \oplus \Z/18\Z\) | 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 | 13 | 89 |
---|---|---|---|---|
Reduction type | split | split | split | nonsplit |
$\lambda$-invariant(s) | 2 | 5 | 1 | 0 |
$\mu$-invariant(s) | 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 $0$.