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Properties

Label 20.20.7548784080...3125.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{35}\cdot 11^{10}$
Root discriminant $55.45$
Ramified primes $5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![51151, 354330, -514350, -2942985, 4443525, 2832749, -5547510, -1129950, 3105870, 230175, -972254, -25110, 184275, 1395, -21600, -31, 1530, 0, -60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 60*x^18 + 1530*x^16 - 31*x^15 - 21600*x^14 + 1395*x^13 + 184275*x^12 - 25110*x^11 - 972254*x^10 + 230175*x^9 + 3105870*x^8 - 1129950*x^7 - 5547510*x^6 + 2832749*x^5 + 4443525*x^4 - 2942985*x^3 - 514350*x^2 + 354330*x + 51151)
 
gp: K = bnfinit(x^20 - 60*x^18 + 1530*x^16 - 31*x^15 - 21600*x^14 + 1395*x^13 + 184275*x^12 - 25110*x^11 - 972254*x^10 + 230175*x^9 + 3105870*x^8 - 1129950*x^7 - 5547510*x^6 + 2832749*x^5 + 4443525*x^4 - 2942985*x^3 - 514350*x^2 + 354330*x + 51151, 1)
 

Normalized defining polynomial

\( x^{20} - 60 x^{18} + 1530 x^{16} - 31 x^{15} - 21600 x^{14} + 1395 x^{13} + 184275 x^{12} - 25110 x^{11} - 972254 x^{10} + 230175 x^{9} + 3105870 x^{8} - 1129950 x^{7} - 5547510 x^{6} + 2832749 x^{5} + 4443525 x^{4} - 2942985 x^{3} - 514350 x^{2} + 354330 x + 51151 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(75487840807181783020496368408203125=5^{35}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.45$
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:  $5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(275=5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{275}(1,·)$, $\chi_{275}(197,·)$, $\chi_{275}(199,·)$, $\chi_{275}(34,·)$, $\chi_{275}(142,·)$, $\chi_{275}(144,·)$, $\chi_{275}(87,·)$, $\chi_{275}(153,·)$, $\chi_{275}(221,·)$, $\chi_{275}(32,·)$, $\chi_{275}(208,·)$, $\chi_{275}(98,·)$, $\chi_{275}(166,·)$, $\chi_{275}(43,·)$, $\chi_{275}(111,·)$, $\chi_{275}(89,·)$, $\chi_{275}(56,·)$, $\chi_{275}(252,·)$, $\chi_{275}(254,·)$, $\chi_{275}(263,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{282001} a^{13} - \frac{68734}{282001} a^{12} - \frac{39}{282001} a^{11} - \frac{63585}{282001} a^{10} + \frac{585}{282001} a^{9} - \frac{128606}{282001} a^{8} - \frac{4212}{282001} a^{7} + \frac{16879}{282001} a^{6} + \frac{14742}{282001} a^{5} + \frac{5403}{282001} a^{4} - \frac{22113}{282001} a^{3} + \frac{58900}{282001} a^{2} + \frac{9477}{282001} a - \frac{103817}{282001}$, $\frac{1}{282001} a^{14} - \frac{42}{282001} a^{12} + \frac{75799}{282001} a^{11} + \frac{693}{282001} a^{10} + \frac{36642}{282001} a^{9} - \frac{5670}{282001} a^{8} + \frac{124298}{282001} a^{7} + \frac{23814}{282001} a^{6} + \frac{52438}{282001} a^{5} - \frac{47628}{282001} a^{4} + \frac{129348}{282001} a^{3} + \frac{35721}{282001} a^{2} - \frac{134009}{282001} a - \frac{4374}{282001}$, $\frac{1}{282001} a^{15} + \frac{8981}{282001} a^{12} - \frac{945}{282001} a^{11} - \frac{95919}{282001} a^{10} + \frac{18900}{282001} a^{9} + \frac{80865}{282001} a^{8} + \frac{128911}{282001} a^{7} - \frac{84647}{282001} a^{6} + \frac{7534}{282001} a^{5} + \frac{74273}{282001} a^{4} - \frac{47022}{282001} a^{3} + \frac{83783}{282001} a^{2} + \frac{111659}{282001} a - \frac{130299}{282001}$, $\frac{1}{282001} a^{16} - \frac{1080}{282001} a^{12} - \frac{27661}{282001} a^{11} + \frac{23760}{282001} a^{10} - \frac{97002}{282001} a^{9} + \frac{63301}{282001} a^{8} - \frac{44809}{282001} a^{7} + \frac{133773}{282001} a^{6} - \frac{65160}{282001} a^{5} - \frac{67193}{282001} a^{4} - \frac{130069}{282001} a^{3} - \frac{117366}{282001} a^{2} - \frac{78934}{282001} a + \frac{85171}{282001}$, $\frac{1}{282001} a^{17} - \frac{94118}{282001} a^{12} - \frac{18360}{282001} a^{11} + \frac{39442}{282001} a^{10} + \frac{131099}{282001} a^{9} + \frac{87204}{282001} a^{8} + \frac{96829}{282001} a^{7} + \frac{116096}{282001} a^{6} + \frac{62111}{282001} a^{5} + \frac{65151}{282001} a^{4} - \frac{29321}{282001} a^{3} + \frac{82841}{282001} a^{2} - \frac{113706}{282001} a + \frac{114038}{282001}$, $\frac{1}{282001} a^{18} - \frac{22032}{282001} a^{12} + \frac{34853}{282001} a^{11} - \frac{18710}{282001} a^{10} - \frac{125962}{282001} a^{9} + \frac{4243}{282001} a^{8} - \frac{97515}{282001} a^{7} - \frac{113801}{282001} a^{6} + \frac{107787}{282001} a^{5} + \frac{42430}{282001} a^{4} + \frac{18887}{282001} a^{3} - \frac{139164}{282001} a^{2} + \frac{101161}{282001} a + \frac{4243}{282001}$, $\frac{1}{282001} a^{19} + \frac{32735}{282001} a^{12} - \frac{31955}{282001} a^{11} - \frac{49714}{282001} a^{10} - \frac{79083}{282001} a^{9} + \frac{1141}{282001} a^{8} - \frac{134256}{282001} a^{7} + \frac{26596}{282001} a^{6} - \frac{26978}{282001} a^{5} + \frac{53361}{282001} a^{4} - \frac{35052}{282001} a^{3} + \frac{17359}{282001} a^{2} + \frac{120767}{282001} a + \frac{13967}{282001}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $19$
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:  \( 73895548051.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.15125.1, 5.5.390625.1, \(\Q(\zeta_{25})^+\)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $20$ $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ R $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$11$11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$