Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+y=x^3+x^2+16211x+856569\) | (homogenize, simplify) |
\(y^2z+yz^2=x^3+x^2z+16211xz^2+856569z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+21009024x+39711984912\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(411, 8781\right)\) |
$\hat{h}(P)$ | ≈ | $2.4618199451010265157600501438$ |
Torsion generators
\( \left(21, 1098\right) \)
Integral points
\( \left(21, 1098\right) \), \( \left(21, -1099\right) \), \( \left(411, 8781\right) \), \( \left(411, -8782\right) \)
Invariants
Conductor: | \( 2093 \) | = | $7 \cdot 13 \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-585612268875179 $ | = | $-1 \cdot 7^{4} \cdot 13^{9} \cdot 23 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{471114356703100928}{585612268875179} \) | = | $2^{21} \cdot 7^{-4} \cdot 13^{-9} \cdot 23^{-1} \cdot 6079^{3}$ | 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: | $1.5197081550887875576626598687\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.5197081550887875576626598687\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0222629824849132\dots$ | |||
Szpiro ratio: | $5.340044565172915\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.4618199451010265157600501438\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.34620175949925302097259358542\dots$ | 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: | $ 36 $ = $ 2^{2}\cdot3^{2}\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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 3.4091455862573194285761702475 $ | 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
$\displaystyle 3.409145586 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.346202 \cdot 2.461820 \cdot 36}{3^2} \approx 3.409145586$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 7776 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$7$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$13$ | $9$ | $I_{9}$ | Split multiplicative | -1 | 1 | 9 | 9 |
$23$ | $1$ | $I_{1}$ | 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 | 9.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 37674 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \cdot 23 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 16381 & 18 \\ 34407 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 11 & 18 \\ 37512 & 20665 \end{array}\right),\left(\begin{array}{rr} 34777 & 18 \\ 11601 & 163 \end{array}\right),\left(\begin{array}{rr} 37657 & 18 \\ 37656 & 19 \end{array}\right),\left(\begin{array}{rr} 10777 & 18 \\ 21078 & 37051 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right)$.
The torsion field $K:=\Q(E[37674])$ is a degree-$2286777243598848$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/37674\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 2093.h
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.1.1196.1 | \(\Z/6\Z\) | Not in database |
$3$ | \(\Q(\zeta_{7})^+\) | \(\Z/9\Z\) | Not in database |
$6$ | 6.0.427694384.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$6$ | 6.0.18141252507.1 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$6$ | 6.0.370229643.3 | \(\Z/9\Z\) | Not in database |
$9$ | 9.3.201271266332864.13 | \(\Z/18\Z\) | Not in database |
$12$ | deg 12 | \(\Z/12\Z\) | Not in database |
$18$ | 18.0.5970377677731885882513770367843.1 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
$18$ | 18.0.62442105498327731433149344118155881185783808.1 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.530749139374986029912275872452429524992.1 | \(\Z/18\Z\) | Not in database |
$18$ | 18.0.1082871997067766282157254127269883904.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | ss | ord | ord | split | ord | split | ord | ord | nonsplit | ord | ord | ord | ord | ord | ss |
$\lambda$-invariant(s) | 2,7 | 1 | 1 | 4 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 |
$\mu$-invariant(s) | 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.