Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-5734x-266524\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-5734xz^2-266524z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-7431291x-12412649898\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Torsion generators
\( \left(140, 1226\right) \)
Integral points
\( \left(140, 1226\right) \), \( \left(140, -1366\right) \)
Invariants
Conductor: | \( 6378 \) | = | $2 \cdot 3 \cdot 1063$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-18511356985344 $ | = | $-1 \cdot 2^{15} \cdot 3^{12} \cdot 1063 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{20849781676074337}{18511356985344} \) | = | $-1 \cdot 2^{-15} \cdot 3^{-12} \cdot 7^{6} \cdot 41^{3} \cdot 137^{3} \cdot 1063^{-1}$ | 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: | $1.2428680658270298454758063415\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.2428680658270298454758063415\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9977437635537839\dots$ | |||
Szpiro ratio: | $4.395355627301559\dots$ |
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: | $0.26453200861956477197985452801\dots$ | 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: | $ 180 $ = $ ( 3 \cdot 5 )\cdot( 2^{2} \cdot 3 )\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: | $3$ | 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) $ ≈ $ 5.2906401723912954395970905602 $ | 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
$\displaystyle 5.290640172 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.264532 \cdot 1.000000 \cdot 180}{3^2} \approx 5.290640172$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 16800 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $15$ | $I_{15}$ | Split multiplicative | -1 | 1 | 15 | 15 |
$3$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
$1063$ | $1$ | $I_{1}$ | 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 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 25512 = 2^{3} \cdot 3 \cdot 1063 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 11694 & 13825 \\ 7441 & 14883 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 19135 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10633 & 6 \\ 6387 & 19 \end{array}\right),\left(\begin{array}{rr} 12757 & 6 \\ 12759 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 25507 & 6 \\ 25506 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[25512])$ is a degree-$5878092234276864$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/25512\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 6378f
consists of 2 curves linked by isogenies of
degree 3.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.1.8504.1 | \(\Z/6\Z\) | Not in database |
$6$ | 6.0.614992408064.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$6$ | 6.0.34474408405947.1 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$9$ | 9.3.1172551300970552296128.1 | \(\Z/9\Z\) | Not in database |
$12$ | deg 12 | \(\Z/12\Z\) | Not in database |
$18$ | 18.0.12136596622084037760068596304394176028139685986172928.1 | \(\Z/3\Z \oplus \Z/6\Z\) | 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 | 1063 |
---|---|---|---|---|
Reduction type | split | split | ord | split |
$\lambda$-invariant(s) | 3 | 1 | 4 | ? |
$\mu$-invariant(s) | 0 | 0 | 0 | ? |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.
An entry ? indicates that the invariants have not yet been computed.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.