↑

Properties

Label 20.0.20776019874...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 3^{10}\cdot 5^{10}$
Root discriminant $26.05$
Ramified primes $2, 3, 5$
Class number $4$
Class group $[2, 2]$
Galois group $F_5$ (as 20T5)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -256, 1872, -4992, 2300, 9872, -6232, 3008, 4276, -1352, 3016, -2032, 1717, -1304, 584, -296, 134, -48, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 24*x^18 - 48*x^17 + 134*x^16 - 296*x^15 + 584*x^14 - 1304*x^13 + 1717*x^12 - 2032*x^11 + 3016*x^10 - 1352*x^9 + 4276*x^8 + 3008*x^7 - 6232*x^6 + 9872*x^5 + 2300*x^4 - 4992*x^3 + 1872*x^2 - 256*x + 16)
 
gp: K = bnfinit(x^20 - 8*x^19 + 24*x^18 - 48*x^17 + 134*x^16 - 296*x^15 + 584*x^14 - 1304*x^13 + 1717*x^12 - 2032*x^11 + 3016*x^10 - 1352*x^9 + 4276*x^8 + 3008*x^7 - 6232*x^6 + 9872*x^5 + 2300*x^4 - 4992*x^3 + 1872*x^2 - 256*x + 16, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 24 x^{18} - 48 x^{17} + 134 x^{16} - 296 x^{15} + 584 x^{14} - 1304 x^{13} + 1717 x^{12} - 2032 x^{11} + 3016 x^{10} - 1352 x^{9} + 4276 x^{8} + 3008 x^{7} - 6232 x^{6} + 9872 x^{5} + 2300 x^{4} - 4992 x^{3} + 1872 x^{2} - 256 x + 16 \)

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:  \(20776019874734407680000000000=2^{55}\cdot 3^{10}\cdot 5^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.05$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{10} - \frac{1}{4} a^{8} + \frac{1}{12} a^{6} + \frac{1}{6} a^{2} + \frac{1}{3}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{11} - \frac{1}{4} a^{9} + \frac{1}{12} a^{7} + \frac{1}{6} a^{3} + \frac{1}{3} a$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{10} - \frac{1}{6} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{12} a^{15} - \frac{1}{12} a^{11} - \frac{1}{6} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{24} a^{16} + \frac{1}{24} a^{8} - \frac{1}{4} a^{4} - \frac{1}{3}$, $\frac{1}{24} a^{17} + \frac{1}{24} a^{9} - \frac{1}{4} a^{5} - \frac{1}{3} a$, $\frac{1}{360} a^{18} + \frac{1}{90} a^{17} - \frac{1}{90} a^{16} - \frac{1}{36} a^{15} - \frac{1}{30} a^{14} - \frac{1}{36} a^{13} - \frac{1}{90} a^{12} + \frac{1}{45} a^{11} + \frac{17}{360} a^{10} - \frac{1}{60} a^{9} - \frac{11}{45} a^{8} + \frac{89}{180} a^{7} - \frac{13}{36} a^{6} - \frac{1}{18} a^{5} - \frac{1}{5} a^{4} + \frac{37}{90} a^{3} + \frac{16}{45} a^{2} + \frac{22}{45} a + \frac{17}{45}$, $\frac{1}{28873894429535141479770861362640} a^{19} - \frac{305963265530218426040690435}{1443694721476757073988543068132} a^{18} - \frac{13375673895238564613458151243}{962463147651171382659028712088} a^{17} + \frac{101157287902735711986303351863}{14436947214767570739885430681320} a^{16} + \frac{182368231125921196117675765189}{14436947214767570739885430681320} a^{15} - \frac{45702051566259064338015180959}{3609236803691892684971357670330} a^{14} - \frac{57286740634204523056518698951}{7218473607383785369942715340660} a^{13} + \frac{17444999904243990505919672659}{1804618401845946342485678835165} a^{12} - \frac{99848494948877697805278347143}{5774778885907028295954172272528} a^{11} + \frac{166939063838452849370325531929}{7218473607383785369942715340660} a^{10} - \frac{2773791893982019622669818906397}{14436947214767570739885430681320} a^{9} + \frac{14856872107772717156301093613}{962463147651171382659028712088} a^{8} - \frac{150613657292457406360201880663}{1203078934563964228323785890110} a^{7} - \frac{12923900603972943725354090596}{40102631152132140944126196337} a^{6} - \frac{58638081692922082825948872632}{1804618401845946342485678835165} a^{5} + \frac{2537176537743202478837810425369}{7218473607383785369942715340660} a^{4} + \frac{1097399561979036739475649538573}{2406157869127928456647571780220} a^{3} + \frac{18877814376800509680162841054}{1804618401845946342485678835165} a^{2} + \frac{358675788203247284112405461753}{1203078934563964228323785890110} a + \frac{768380922239711383876946905636}{1804618401845946342485678835165}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6532127.80174 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.460800.2, 5.1.460800.1 x5, 10.2.1698693120000.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.460800.1
Degree 10 sibling: 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 R R ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$