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Properties

Label 20.20.2372891715...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{8}\cdot 5^{24}\cdot 41^{15}$
Root discriminant $147.49$
Ramified primes $2, 5, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5:D_5.Q_8$ (as 20T105)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-96466096, 788973840, -594572400, -1268735640, 1247950800, 573003188, -722962260, -113724000, 199389060, 11583000, -31122351, -631800, 2948400, 17550, -172800, -195, 6120, 0, -120, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 120*x^18 + 6120*x^16 - 195*x^15 - 172800*x^14 + 17550*x^13 + 2948400*x^12 - 631800*x^11 - 31122351*x^10 + 11583000*x^9 + 199389060*x^8 - 113724000*x^7 - 722962260*x^6 + 573003188*x^5 + 1247950800*x^4 - 1268735640*x^3 - 594572400*x^2 + 788973840*x - 96466096)
 
gp: K = bnfinit(x^20 - 120*x^18 + 6120*x^16 - 195*x^15 - 172800*x^14 + 17550*x^13 + 2948400*x^12 - 631800*x^11 - 31122351*x^10 + 11583000*x^9 + 199389060*x^8 - 113724000*x^7 - 722962260*x^6 + 573003188*x^5 + 1247950800*x^4 - 1268735640*x^3 - 594572400*x^2 + 788973840*x - 96466096, 1)
 

Normalized defining polynomial

\( x^{20} - 120 x^{18} + 6120 x^{16} - 195 x^{15} - 172800 x^{14} + 17550 x^{13} + 2948400 x^{12} - 631800 x^{11} - 31122351 x^{10} + 11583000 x^{9} + 199389060 x^{8} - 113724000 x^{7} - 722962260 x^{6} + 573003188 x^{5} + 1247950800 x^{4} - 1268735640 x^{3} - 594572400 x^{2} + 788973840 x - 96466096 \)

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:  \(23728917159905058454721084594726562500000000=2^{8}\cdot 5^{24}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $147.49$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{32392640900} a^{15} - \frac{9}{3239264090} a^{13} + \frac{162}{1619632045} a^{11} - \frac{5166429967}{32392640900} a^{10} - \frac{594}{323926409} a^{9} - \frac{697030549}{1619632045} a^{8} + \frac{5832}{323926409} a^{7} + \frac{60953124}{1619632045} a^{6} - \frac{3096059127}{32392640900} a^{5} - \frac{150766271}{323926409} a^{4} - \frac{437779689}{3239264090} a^{3} + \frac{128372404}{323926409} a^{2} - \frac{304543378}{1619632045} a - \frac{1773563192}{8098160225}$, $\frac{1}{32392640900} a^{16} - \frac{9}{3239264090} a^{14} + \frac{162}{1619632045} a^{12} - \frac{5166429967}{32392640900} a^{11} - \frac{594}{323926409} a^{10} - \frac{697030549}{1619632045} a^{9} + \frac{5832}{323926409} a^{8} + \frac{60953124}{1619632045} a^{7} - \frac{3096059127}{32392640900} a^{6} - \frac{150766271}{323926409} a^{5} - \frac{437779689}{3239264090} a^{4} + \frac{128372404}{323926409} a^{3} - \frac{304543378}{1619632045} a^{2} - \frac{1773563192}{8098160225} a$, $\frac{1}{32392640900} a^{17} - \frac{243}{1619632045} a^{13} - \frac{5166429967}{32392640900} a^{12} + \frac{2322}{323926409} a^{11} + \frac{697030549}{3239264090} a^{10} - \frac{47628}{323926409} a^{9} + \frac{493853469}{1619632045} a^{8} - \frac{3043571127}{32392640900} a^{7} - \frac{25389266}{323926409} a^{6} + \frac{425532489}{1619632045} a^{5} - \frac{159609217}{323926409} a^{4} - \frac{569044843}{1619632045} a^{3} + \frac{3628737933}{8098160225} a^{2} + \frac{24968149}{323926409} a + \frac{468503444}{1619632045}$, $\frac{1}{37245123292101800} a^{18} - \frac{21101}{1862256164605090} a^{17} - \frac{27}{9311280823025450} a^{16} + \frac{251683}{18622561646050900} a^{15} + \frac{243}{1862256164605090} a^{14} - \frac{6005210071884907}{37245123292101800} a^{13} + \frac{621585031273021}{1862256164605090} a^{12} - \frac{7890078553363267}{18622561646050900} a^{11} - \frac{2438239976114761}{18622561646050900} a^{10} - \frac{539140018137829}{1862256164605090} a^{9} - \frac{2338313467050507}{37245123292101800} a^{8} - \frac{134880566300517}{372451232921018} a^{7} - \frac{634766015855603}{4655640411512725} a^{6} - \frac{6293266799985441}{18622561646050900} a^{5} - \frac{62350245407441}{931128082302545} a^{4} + \frac{1313887378468173}{9311280823025450} a^{3} - \frac{225801629709719}{931128082302545} a^{2} + \frac{74404389404983}{4655640411512725} a - \frac{1666484069930411}{4655640411512725}$, $\frac{1}{37245123292101800} a^{19} - \frac{57}{18622561646050900} a^{17} - \frac{58129}{9311280823025450} a^{16} + \frac{684}{4655640411512725} a^{15} - \frac{6005212056672727}{37245123292101800} a^{14} - \frac{3591}{931128082302545} a^{13} - \frac{137047680035863}{300363897516950} a^{12} - \frac{2945196461581437}{9311280823025450} a^{11} + \frac{530561569016019}{9311280823025450} a^{10} + \frac{3961856714549853}{37245123292101800} a^{9} - \frac{360301353442032}{931128082302545} a^{8} - \frac{277513674887781}{600727795033900} a^{7} + \frac{4392052940456373}{9311280823025450} a^{6} - \frac{3515848987697411}{9311280823025450} a^{5} + \frac{1998613002842009}{4655640411512725} a^{4} + \frac{459650300609349}{931128082302545} a^{3} - \frac{1605013478916676}{4655640411512725} a^{2} - \frac{938077260903429}{4655640411512725} a + \frac{606431665456013}{4655640411512725}$

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:  \( 4001313805280000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:D_5.Q_8$ (as 20T105):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $C_5:D_5.Q_8$
Character table for $C_5:D_5.Q_8$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 10.10.452563285156250000.1

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

Sibling fields

Degree 20 sibling: 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 $20$ $20$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.10.14.8$x^{10} + 5 x^{8} + 20 x^{6} + 10 x^{5} + 5 x^{4} + 10 x^{2} + 12$$5$$2$$14$$F_{5}\times C_2$$[7/4]_{4}^{2}$
5.10.10.7$x^{10} + 10 x^{8} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 12$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
41Data not computed