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Properties

Label 20.0.69654621282...6096.5
Degree $20$
Signature $[0, 10]$
Discriminant $2^{10}\cdot 11^{16}\cdot 23^{6}$
Root discriminant $24.67$
Ramified primes $2, 11, 23$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T751

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![593, 686, 1338, 554, 2997, -3668, 8050, -10163, 11981, -11136, 8623, -5379, 2492, -676, -44, 191, -96, 26, 1, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + x^18 + 26*x^17 - 96*x^16 + 191*x^15 - 44*x^14 - 676*x^13 + 2492*x^12 - 5379*x^11 + 8623*x^10 - 11136*x^9 + 11981*x^8 - 10163*x^7 + 8050*x^6 - 3668*x^5 + 2997*x^4 + 554*x^3 + 1338*x^2 + 686*x + 593)
 
gp: K = bnfinit(x^20 - 3*x^19 + x^18 + 26*x^17 - 96*x^16 + 191*x^15 - 44*x^14 - 676*x^13 + 2492*x^12 - 5379*x^11 + 8623*x^10 - 11136*x^9 + 11981*x^8 - 10163*x^7 + 8050*x^6 - 3668*x^5 + 2997*x^4 + 554*x^3 + 1338*x^2 + 686*x + 593, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + x^{18} + 26 x^{17} - 96 x^{16} + 191 x^{15} - 44 x^{14} - 676 x^{13} + 2492 x^{12} - 5379 x^{11} + 8623 x^{10} - 11136 x^{9} + 11981 x^{8} - 10163 x^{7} + 8050 x^{6} - 3668 x^{5} + 2997 x^{4} + 554 x^{3} + 1338 x^{2} + 686 x + 593 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6965462128295683654948996096=2^{10}\cdot 11^{16}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.67$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 11, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} + \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{132} a^{15} + \frac{7}{132} a^{14} + \frac{5}{33} a^{13} - \frac{29}{132} a^{12} - \frac{35}{132} a^{11} + \frac{5}{132} a^{10} - \frac{23}{66} a^{9} - \frac{43}{132} a^{8} + \frac{1}{44} a^{7} - \frac{10}{33} a^{6} - \frac{1}{12} a^{5} - \frac{23}{66} a^{4} - \frac{49}{132} a^{3} + \frac{5}{11} a^{2} + \frac{29}{132} a - \frac{17}{66}$, $\frac{1}{264} a^{16} - \frac{1}{264} a^{15} - \frac{1}{88} a^{14} - \frac{1}{11} a^{13} - \frac{1}{264} a^{12} - \frac{1}{22} a^{11} + \frac{13}{264} a^{10} + \frac{7}{66} a^{9} - \frac{49}{264} a^{8} - \frac{31}{264} a^{7} - \frac{9}{44} a^{6} + \frac{9}{22} a^{5} + \frac{1}{3} a^{4} - \frac{19}{66} a^{3} - \frac{1}{3} a^{2} + \frac{8}{33} a - \frac{91}{264}$, $\frac{1}{264} a^{17} + \frac{1}{264} a^{14} + \frac{5}{24} a^{13} + \frac{1}{88} a^{12} - \frac{7}{264} a^{11} - \frac{71}{264} a^{10} + \frac{59}{264} a^{9} + \frac{1}{22} a^{8} + \frac{59}{264} a^{7} - \frac{53}{132} a^{6} - \frac{14}{33} a^{5} - \frac{5}{33} a^{4} - \frac{4}{11} a^{3} - \frac{2}{11} a^{2} + \frac{89}{264} a - \frac{95}{264}$, $\frac{1}{528} a^{18} - \frac{1}{528} a^{17} + \frac{1}{528} a^{15} + \frac{9}{88} a^{14} + \frac{5}{33} a^{13} - \frac{5}{264} a^{12} + \frac{25}{66} a^{11} - \frac{1}{264} a^{10} + \frac{217}{528} a^{9} - \frac{85}{528} a^{8} - \frac{1}{16} a^{7} - \frac{1}{88} a^{6} + \frac{17}{44} a^{5} - \frac{47}{132} a^{4} - \frac{9}{22} a^{3} - \frac{127}{528} a^{2} - \frac{13}{132} a - \frac{37}{528}$, $\frac{1}{4972761962426689376374704} a^{19} - \frac{1756900621813372218397}{4972761962426689376374704} a^{18} + \frac{3700035648008268281101}{2486380981213344688187352} a^{17} - \frac{5149046532842034999463}{4972761962426689376374704} a^{16} - \frac{3975346316198579732743}{2486380981213344688187352} a^{15} - \frac{96170595876005617631409}{828793660404448229395784} a^{14} - \frac{36276972566139253956769}{1243190490606672344093676} a^{13} + \frac{6310178243553727896537}{75344878218586202672344} a^{12} - \frac{221068995892841918739193}{621595245303336172046838} a^{11} + \frac{1347401716014604327024331}{4972761962426689376374704} a^{10} - \frac{1028195340717697906806643}{4972761962426689376374704} a^{9} - \frac{800506008273890354425103}{1657587320808896458791568} a^{8} - \frac{121222007781309967402841}{414396830202224114697892} a^{7} + \frac{127625889337332532320017}{1243190490606672344093676} a^{6} - \frac{64439136823251155518079}{1243190490606672344093676} a^{5} + \frac{1272812638215023184243}{414396830202224114697892} a^{4} - \frac{2123629923376646836891759}{4972761962426689376374704} a^{3} - \frac{299909031219295425277703}{1243190490606672344093676} a^{2} - \frac{189374213134844165463031}{4972761962426689376374704} a + \frac{445573335999061464290815}{2486380981213344688187352}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 152173.593776 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T751:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n751 are not computed
Character table for t20n751 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.4.4930254263.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.10.10.5$x^{10} - 9 x^{8} + 50 x^{6} - 50 x^{4} + 45 x^{2} - 5$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$