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Properties

Label 20.0.18444893486...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{10}\cdot 11^{5}\cdot 5783^{4}$
Root discriminant $46.06$
Ramified primes $2, 5, 11, 5783$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 20T174

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1126631, -3157754, 5034175, -6044510, 6184004, -5414292, 4222990, -2909058, 1852214, -1047220, 553307, -258210, 113629, -43334, 15650, -4724, 1357, -300, 63, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 63*x^18 - 300*x^17 + 1357*x^16 - 4724*x^15 + 15650*x^14 - 43334*x^13 + 113629*x^12 - 258210*x^11 + 553307*x^10 - 1047220*x^9 + 1852214*x^8 - 2909058*x^7 + 4222990*x^6 - 5414292*x^5 + 6184004*x^4 - 6044510*x^3 + 5034175*x^2 - 3157754*x + 1126631)
 
gp: K = bnfinit(x^20 - 8*x^19 + 63*x^18 - 300*x^17 + 1357*x^16 - 4724*x^15 + 15650*x^14 - 43334*x^13 + 113629*x^12 - 258210*x^11 + 553307*x^10 - 1047220*x^9 + 1852214*x^8 - 2909058*x^7 + 4222990*x^6 - 5414292*x^5 + 6184004*x^4 - 6044510*x^3 + 5034175*x^2 - 3157754*x + 1126631, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 63 x^{18} - 300 x^{17} + 1357 x^{16} - 4724 x^{15} + 15650 x^{14} - 43334 x^{13} + 113629 x^{12} - 258210 x^{11} + 553307 x^{10} - 1047220 x^{9} + 1852214 x^{8} - 2909058 x^{7} + 4222990 x^{6} - 5414292 x^{5} + 6184004 x^{4} - 6044510 x^{3} + 5034175 x^{2} - 3157754 x + 1126631 \)

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:  \(1844489348608657807063040000000000=2^{20}\cdot 5^{10}\cdot 11^{5}\cdot 5783^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.06$
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:  $2, 5, 11, 5783$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{4046526661671911} a^{18} - \frac{930554385883112}{4046526661671911} a^{17} + \frac{28153261683520}{76349559654187} a^{16} + \frac{615514032360843}{4046526661671911} a^{15} + \frac{228458996504944}{4046526661671911} a^{14} + \frac{806974331667197}{4046526661671911} a^{13} - \frac{886622239691892}{4046526661671911} a^{12} + \frac{720398128693113}{4046526661671911} a^{11} + \frac{234164965213559}{4046526661671911} a^{10} - \frac{1030878556156569}{4046526661671911} a^{9} + \frac{459356941396964}{4046526661671911} a^{8} + \frac{137483170521506}{4046526661671911} a^{7} - \frac{955661982193993}{4046526661671911} a^{6} + \frac{299254678314147}{4046526661671911} a^{5} + \frac{594447390208956}{4046526661671911} a^{4} - \frac{1512435462676294}{4046526661671911} a^{3} - \frac{124249118978748}{4046526661671911} a^{2} + \frac{1603622946474306}{4046526661671911} a - \frac{811395971973644}{4046526661671911}$, $\frac{1}{236130459132010190911847742215793151979} a^{19} - \frac{14659838961219917572588}{236130459132010190911847742215793151979} a^{18} - \frac{84969906703743202136599912046403839250}{236130459132010190911847742215793151979} a^{17} + \frac{14192790648371340729260130443224582556}{236130459132010190911847742215793151979} a^{16} + \frac{110102734805828283663522595158926397148}{236130459132010190911847742215793151979} a^{15} + \frac{99327455026703650718382815698124824525}{236130459132010190911847742215793151979} a^{14} + \frac{7398066013461093121120735100985353750}{236130459132010190911847742215793151979} a^{13} + \frac{63419556433582225410106762530926606420}{236130459132010190911847742215793151979} a^{12} - \frac{28665036744015337890417521356202753790}{236130459132010190911847742215793151979} a^{11} + \frac{44429860899553873020015013970284687831}{236130459132010190911847742215793151979} a^{10} - \frac{29486610774869776817969403528992992126}{236130459132010190911847742215793151979} a^{9} - \frac{102990810495860745695294405997797921519}{236130459132010190911847742215793151979} a^{8} - \frac{111538789963545237038691609874820286298}{236130459132010190911847742215793151979} a^{7} + \frac{82220764401794378576021994362393684659}{236130459132010190911847742215793151979} a^{6} + \frac{62548581176569260451927746402979533928}{236130459132010190911847742215793151979} a^{5} + \frac{85760025794045939832911207257304487367}{236130459132010190911847742215793151979} a^{4} + \frac{331946536194743062579492753844087169}{236130459132010190911847742215793151979} a^{3} + \frac{7352508769001192458402310609133906794}{236130459132010190911847742215793151979} a^{2} - \frac{34203738099390496605602664252926441269}{236130459132010190911847742215793151979} a + \frac{38402820142398504832213585319885915724}{236130459132010190911847742215793151979}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 55192072.3215 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T174:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 960
The 35 conjugacy class representatives for t20n174
Character table for t20n174 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.4400.1, 5.3.5783.1, 10.6.104509653125.1

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
Degree 24 siblings: data not computed
Degree 40 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 $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5783Data not computed