Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2-20584x-1129879\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z-20584xz^2-1129879z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-26676891x-52315472970\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{4}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-77, 115\right)\) |
$\hat{h}(P)$ | ≈ | $2.5145235462924666746586243578$ |
Torsion generators
\( \left(-91, 45\right) \), \( \left(-83, -83\right) \)
Integral points
\( \left(-91, 45\right) \), \( \left(-83, 165\right) \), \( \left(-83, -83\right) \), \( \left(-77, 115\right) \), \( \left(-77, -39\right) \), \( \left(165, -83\right) \), \( \left(197, 1485\right) \), \( \left(197, -1683\right) \), \( \left(413, 7605\right) \), \( \left(413, -8019\right) \), \( \left(1281, 44929\right) \), \( \left(1281, -46211\right) \), \( \left(2087, 94095\right) \), \( \left(2087, -96183\right) \)
Invariants
Conductor: | \( 1302 \) | = | $2 \cdot 3 \cdot 7 \cdot 31$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $15013703061504 $ | = | $2^{12} \cdot 3^{4} \cdot 7^{2} \cdot 31^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{964526913483831937}{15013703061504} \) | = | $2^{-12} \cdot 3^{-4} \cdot 7^{-2} \cdot 31^{-4} \cdot 988033^{3}$ | 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: | $1.3293758380515405432772684570\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.3293758380515405432772684570\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9938490923577851\dots$ | |||
Szpiro ratio: | $5.774176756405347\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.5145235462924666746586243578\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.39895753203485839219510077467\dots$ | 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: | $ 192 $ = $ ( 2^{2} \cdot 3 )\cdot2\cdot2\cdot2^{2} $ | 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: | $8$ | 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) $ ≈ $ 3.0095643248171475080235904743 $ | 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
$\displaystyle 3.009564325 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.398958 \cdot 2.514524 \cdot 192}{8^2} \approx 3.009564325$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 4608 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
$3$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$7$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$31$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
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 |
---|---|---|
$2$ | 2Cs | 8.48.0.25 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1736 = 2^{3} \cdot 7 \cdot 31 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 7 & 6 \\ 738 & 1731 \end{array}\right),\left(\begin{array}{rr} 873 & 440 \\ 408 & 837 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 1732 & 1733 \end{array}\right),\left(\begin{array}{rr} 5 & 1308 \\ 1730 & 427 \end{array}\right),\left(\begin{array}{rr} 1729 & 8 \\ 1728 & 9 \end{array}\right),\left(\begin{array}{rr} 561 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1736])$ is a degree-$14399078400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1736\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 1302.j
consists of 6 curves linked by isogenies of
degrees dividing 8.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{4}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$4$ | \(\Q(i, \sqrt{7})\) | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{2}, \sqrt{-31})\) | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$4$ | \(\Q(\sqrt{14}, \sqrt{62})\) | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.0.2439569664.6 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.2.392801137983387.3 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$16$ | 16.0.21117268020957937382075662336.1 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | 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 | split | nonsplit | ord | nonsplit | ord | ord | ord | ord | ss | ord | split | ord | ord | ord | ss |
$\lambda$-invariant(s) | 9 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 | 1 | 2 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.