Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-1080971082x+13678914712356\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-1080971082xz^2+13678914712356z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1400938522299x+638207647635248406\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{9}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(53556, 10432614\right)\) |
$\hat{h}(P)$ | ≈ | $5.5220765263564171379346219972$ |
Torsion generators
\( \left(19116, 14094\right) \)
Integral points
\( \left(19116, 14094\right) \), \( \left(19116, -33210\right) \), \( \left(19764, 176742\right) \), \( \left(19764, -196506\right) \), \( \left(24372, 1333350\right) \), \( \left(24372, -1357722\right) \), \( \left(53556, 10432614\right) \), \( \left(53556, -10486170\right) \), \( \left(66420, 15293286\right) \), \( \left(66420, -15359706\right) \), \( \left(85692, 23415558\right) \), \( \left(85692, -23501250\right) \), \( \left(3898044, 7693867782\right) \), \( \left(3898044, -7697765826\right) \)
Invariants
Conductor: | \( 118698 \) | = | $2 \cdot 3 \cdot 73 \cdot 271$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $5481890282879036260614144 $ | = | $2^{27} \cdot 3^{18} \cdot 73^{3} \cdot 271 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{139690200244171980376257072833953}{5481890282879036260614144} \) | = | $2^{-27} \cdot 3^{-18} \cdot 73^{-3} \cdot 271^{-1} \cdot 487^{3} \cdot 106543351^{3}$ | 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: | $3.8309642572691443487428792867\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $3.8309642572691443487428792867\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0088851417385944\dots$ | |||
Szpiro ratio: | $6.334717602954133\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $5.5220765263564171379346219972\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.071462739121647836084935623550\dots$ | 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: | $ 1458 $ = $ 3^{3}\cdot( 2 \cdot 3^{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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 7.1032088558301105425404502420 $ | 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
$\displaystyle 7.103208856 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.071463 \cdot 5.522077 \cdot 1458}{9^2} \approx 7.103208856$
Modular invariants
Modular form 118698.2.a.f
For more coefficients, see the Downloads section to the right.
Modular degree: | 69004224 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $27$ | $I_{27}$ | Split multiplicative | -1 | 1 | 27 | 27 |
$3$ | $18$ | $I_{18}$ | Split multiplicative | -1 | 1 | 18 | 18 |
$73$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$271$ | $1$ | $I_{1}$ | 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 \( 1424376 = 2^{3} \cdot 3^{2} \cdot 73 \cdot 271 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 356105 & 712206 \\ 712026 & 988885 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 1424359 & 18 \\ 1424358 & 19 \end{array}\right),\left(\begin{array}{rr} 546337 & 18 \\ 643905 & 163 \end{array}\right),\left(\begin{array}{rr} 998641 & 18 \\ 441513 & 163 \end{array}\right),\left(\begin{array}{rr} 356095 & 18 \\ 356103 & 163 \end{array}\right),\left(\begin{array}{rr} 712189 & 18 \\ 712197 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right)$.
The torsion field $K:=\Q(E[1424376])$ is a degree-$6240800156657437900800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1424376\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 118698i
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 |
---|---|---|---|
$3$ | 3.3.158264.1 | \(\Z/18\Z\) | Not in database |
$6$ | 6.6.3964116542303744.1 | \(\Z/2\Z \oplus \Z/18\Z\) | Not in database |
$6$ | 6.0.145626672987.1 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
$9$ | 9.3.11712056128402461713779895163.1 | \(\Z/27\Z\) | Not in database |
$12$ | deg 12 | \(\Z/36\Z\) | Not in database |
$18$ | 18.0.8997843752892634480763249400634258159365566607536160768.1 | \(\Z/3\Z \oplus \Z/18\Z\) | 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 | 73 | 271 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | split | ord | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord | ord | split | split |
$\lambda$-invariant(s) | 9 | 6 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
$\mu$-invariant(s) | 0 | 0 | 0 | 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.