Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-193010459x+1031970436427\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-193010459xz^2+1031970436427z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-3088167339x+66043019764006\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{9}\Z\)
Torsion generators
\( \left(8139, 8698\right) \)
Integral points
\( \left(8139, 8698\right) \), \( \left(8139, -16838\right) \), \( \left(8587, 83514\right) \), \( \left(8587, -92102\right) \), \( \left(11331, 541762\right) \), \( \left(11331, -553094\right) \), \( \left(33675, 5703226\right) \), \( \left(33675, -5736902\right) \)
Invariants
Conductor: | \( 122094 \) | = | $2 \cdot 3^{3} \cdot 7 \cdot 17 \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $153464610297375551913984 $ | = | $2^{27} \cdot 3^{5} \cdot 7^{9} \cdot 17 \cdot 19^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{3272351648557113352874597691}{631541606162039308288} \) | = | $2^{-27} \cdot 3 \cdot 7^{-9} \cdot 17^{-1} \cdot 19^{-3} \cdot 107^{3} \cdot 307^{3} \cdot 31337^{3}$ | 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);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $3.4492707380650007855084024877\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.9915156177866217474270503057\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0140231326610487\dots$ | |||
Szpiro ratio: | $5.878172413526001\dots$ |
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: | $0.099667168775974204017073532402\dots$ | 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: | $ 2187 $ = $ 3^{3}\cdot3\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) $ ≈ $ 2.6910135569513035084609853749 $ | 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
$\displaystyle 2.691013557 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.099667 \cdot 1.000000 \cdot 2187}{9^2} \approx 2.691013557$
Modular invariants
Modular form 122094.2.a.be
For more coefficients, see the Downloads section to the right.
Modular degree: | 31492800 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $27$ | $I_{27}$ | Split multiplicative | -1 | 1 | 27 | 27 |
$3$ | $3$ | $IV$ | Additive | 1 | 3 | 5 | 0 |
$7$ | $9$ | $I_{9}$ | Split multiplicative | -1 | 1 | 9 | 9 |
$17$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$19$ | $3$ | $I_{3}$ | 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 \( 162792 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 17 \cdot 19 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 81397 & 18 \\ 81405 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 122105 & 81414 \\ 41031 & 77419 \end{array}\right),\left(\begin{array}{rr} 23257 & 18 \\ 46521 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 134065 & 18 \\ 67041 & 163 \end{array}\right),\left(\begin{array}{rr} 162775 & 18 \\ 162774 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 154225 & 18 \\ 85689 & 163 \end{array}\right),\left(\begin{array}{rr} 122095 & 18 \\ 122103 & 163 \end{array}\right)$.
The torsion field $K:=\Q(E[162792])$ is a degree-$806372124540272640$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/162792\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 122094bl
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 122094x3, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{9}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.3.488376.1 | \(\Z/18\Z\) | Not in database |
$6$ | 6.6.12942567273291264.1 | \(\Z/2\Z \oplus \Z/18\Z\) | Not in database |
$6$ | 6.0.14795494587.3 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
$9$ | 9.3.537290758872798743315572083.1 | \(\Z/27\Z\) | Not in database |
$12$ | deg 12 | \(\Z/36\Z\) | Not in database |
$18$ | 18.0.110007195162375897181172420727265548102974191845507072.1 | \(\Z/3\Z \oplus \Z/18\Z\) | 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 | 7 | 17 | 19 |
---|---|---|---|---|---|
Reduction type | split | add | split | nonsplit | split |
$\lambda$-invariant(s) | 2 | - | 1 | 0 | 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 $0$.