Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+x^2-41x-199\) | (homogenize, simplify) |
\(y^2z=x^3+x^2z-41xz^2-199z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-3348x-135054\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(8, 7\right)\) |
$\hat{h}(P)$ | ≈ | $2.5360150476328512380934459508$ |
Integral points
\((8,\pm 7)\)
Invariants
Conductor: | \( 704 \) | = | $2^{6} \cdot 11$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-10307264 $ | = | $-1 \cdot 2^{6} \cdot 11^{5} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{122023936}{161051} \) | = | $-1 \cdot 2^{12} \cdot 11^{-5} \cdot 31^{3}$ | 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: | $0.038563749161569590019601799271\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.30800984111840306468901426146\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.012998630378065\dots$ | |||
Szpiro ratio: | $3.6552490760901506\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.5360150476328512380934459508\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.89746650580113239778776783382\dots$ | 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: | $ 1 $ | 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: | $1$ | 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) $ ≈ $ 2.2759885634581473397726141698 $ | 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
$\displaystyle 2.275988563 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.897467 \cdot 2.536015 \cdot 1}{1^2} \approx 2.275988563$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 80 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $II$ | Additive | 1 | 6 | 6 | 0 |
$11$ | $1$ | $I_{5}$ | Non-split multiplicative | 1 | 1 | 5 | 5 |
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 |
---|---|---|
$5$ | 5Cs.4.1 | 5.60.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \), index $1200$, genus $37$, and generators
$\left(\begin{array}{rr} 551 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 10 & 501 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 101 & 50 \\ 204 & 277 \end{array}\right),\left(\begin{array}{rr} 1099 & 0 \\ 0 & 2199 \end{array}\right),\left(\begin{array}{rr} 16 & 35 \\ 115 & 11 \end{array}\right),\left(\begin{array}{rr} 2151 & 50 \\ 2150 & 51 \end{array}\right),\left(\begin{array}{rr} 56 & 25 \\ 135 & 396 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 50 & 1 \end{array}\right),\left(\begin{array}{rr} 16 & 35 \\ 1765 & 1661 \end{array}\right),\left(\begin{array}{rr} 501 & 50 \\ 500 & 51 \end{array}\right)$.
The torsion field $K:=\Q(E[2200])$ is a degree-$5068800000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2200\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 704.h
consists of 3 curves linked by isogenies of
degrees dividing 25.
Twists
The minimal quadratic twist of this elliptic curve is 11.a2, its twist by $8$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{2}) \) | \(\Z/5\Z\) | 2.2.8.1-121.1-a2 |
$3$ | 3.1.44.1 | \(\Z/2\Z\) | Not in database |
$4$ | 4.0.8000.2 | \(\Z/5\Z\) | Not in database |
$6$ | 6.0.21296.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.2.247808.1 | \(\Z/10\Z\) | Not in database |
$8$ | 8.2.131153375232.11 | \(\Z/3\Z\) | Not in database |
$8$ | 8.0.64000000.2 | \(\Z/5\Z \oplus \Z/5\Z\) | Not in database |
$12$ | 12.2.1307761908383744.2 | \(\Z/4\Z\) | Not in database |
$12$ | deg 12 | \(\Z/10\Z\) | Not in database |
$12$ | 12.0.7430465388544.6 | \(\Z/2\Z \oplus \Z/10\Z\) | Not in database |
$16$ | deg 16 | \(\Z/15\Z\) | 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 | add | ord | ord | ord | nonsplit | ord | ord | ss | ord | ss | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | - | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | - | 0 | 1 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.