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Properties

Label 20.0.73038164166...9296.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 11^{16}\cdot 23^{6}$
Root discriminant $49.34$
Ramified primes $2, 11, 23$
Class number $160$ (GRH)
Class group $[2, 4, 20]$ (GRH)
Galois group $C_2\times C_2^4:C_5$ (as 20T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![805751, 611294, 269144, -133254, -72149, -428398, 398012, -235632, 336647, -178938, 108170, -73808, 33069, -11542, 6360, -1196, 590, -82, 36, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 36*x^18 - 82*x^17 + 590*x^16 - 1196*x^15 + 6360*x^14 - 11542*x^13 + 33069*x^12 - 73808*x^11 + 108170*x^10 - 178938*x^9 + 336647*x^8 - 235632*x^7 + 398012*x^6 - 428398*x^5 - 72149*x^4 - 133254*x^3 + 269144*x^2 + 611294*x + 805751)
 
gp: K = bnfinit(x^20 - 4*x^19 + 36*x^18 - 82*x^17 + 590*x^16 - 1196*x^15 + 6360*x^14 - 11542*x^13 + 33069*x^12 - 73808*x^11 + 108170*x^10 - 178938*x^9 + 336647*x^8 - 235632*x^7 + 398012*x^6 - 428398*x^5 - 72149*x^4 - 133254*x^3 + 269144*x^2 + 611294*x + 805751, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 36 x^{18} - 82 x^{17} + 590 x^{16} - 1196 x^{15} + 6360 x^{14} - 11542 x^{13} + 33069 x^{12} - 73808 x^{11} + 108170 x^{10} - 178938 x^{9} + 336647 x^{8} - 235632 x^{7} + 398012 x^{6} - 428398 x^{5} - 72149 x^{4} - 133254 x^{3} + 269144 x^{2} + 611294 x + 805751 \)

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:  \(7303816416639774784171798530359296=2^{30}\cdot 11^{16}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.34$
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 a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23} a^{17} + \frac{1}{23} a^{16} - \frac{3}{23} a^{15} - \frac{2}{23} a^{14} - \frac{7}{23} a^{12} - \frac{2}{23} a^{11} + \frac{3}{23} a^{10} - \frac{1}{23} a^{9} - \frac{2}{23} a^{8} - \frac{8}{23} a^{7} + \frac{3}{23} a^{6} - \frac{7}{23} a^{5} - \frac{11}{23} a^{4} - \frac{6}{23} a^{2} + \frac{7}{23} a + \frac{8}{23}$, $\frac{1}{23} a^{18} - \frac{4}{23} a^{16} + \frac{1}{23} a^{15} + \frac{2}{23} a^{14} - \frac{7}{23} a^{13} + \frac{5}{23} a^{12} + \frac{5}{23} a^{11} - \frac{4}{23} a^{10} - \frac{1}{23} a^{9} - \frac{6}{23} a^{8} + \frac{11}{23} a^{7} - \frac{10}{23} a^{6} - \frac{4}{23} a^{5} + \frac{11}{23} a^{4} - \frac{6}{23} a^{3} - \frac{10}{23} a^{2} + \frac{1}{23} a - \frac{8}{23}$, $\frac{1}{4804049004628136761467451497915135586300439962085225710851838193} a^{19} + \frac{1515845333590654068838566096063096131760567581478505722573913}{208871695853397250498584847735440677665236520090661987428340791} a^{18} - \frac{59703020125644306581611160733042301594731849566822732144984657}{4804049004628136761467451497915135586300439962085225710851838193} a^{17} - \frac{382040471839805698600230425417343722175494982637015834359448990}{4804049004628136761467451497915135586300439962085225710851838193} a^{16} - \frac{2274541613557574651543023992977457995690588039718492259281682409}{4804049004628136761467451497915135586300439962085225710851838193} a^{15} + \frac{1366401325222057310349692973331156250512913248020348735535494412}{4804049004628136761467451497915135586300439962085225710851838193} a^{14} + \frac{1689574488831155650859300296223844897778543418583569330888249863}{4804049004628136761467451497915135586300439962085225710851838193} a^{13} + \frac{238498977730879164956214245733261578670717915865175187710215916}{4804049004628136761467451497915135586300439962085225710851838193} a^{12} - \frac{2388918969334970126819474797582089881677556489740963428731302020}{4804049004628136761467451497915135586300439962085225710851838193} a^{11} - \frac{1829098540533603000855904508788347012281657376021326945309785033}{4804049004628136761467451497915135586300439962085225710851838193} a^{10} + \frac{146016574967244724009967573006745386888752230611779253727584973}{4804049004628136761467451497915135586300439962085225710851838193} a^{9} - \frac{1927366254885705865298133043132220309003736355649372357969990574}{4804049004628136761467451497915135586300439962085225710851838193} a^{8} - \frac{1042342478073510522343215242452581924380347748472085769546968271}{4804049004628136761467451497915135586300439962085225710851838193} a^{7} + \frac{787522202883549130359734015061915516338027399631240145168675601}{4804049004628136761467451497915135586300439962085225710851838193} a^{6} - \frac{1461439645806447331599163285864613197181556470422939856224273103}{4804049004628136761467451497915135586300439962085225710851838193} a^{5} + \frac{105475629966910310490957459707092391403263103119305548472163511}{4804049004628136761467451497915135586300439962085225710851838193} a^{4} - \frac{890074301743887415111558629151889916850891605825806678269509758}{4804049004628136761467451497915135586300439962085225710851838193} a^{3} + \frac{195315371067353170385306268950333945846969319321955872616676048}{4804049004628136761467451497915135586300439962085225710851838193} a^{2} + \frac{596756966374199774739709602600082340097348102463202992733825481}{4804049004628136761467451497915135586300439962085225710851838193} a + \frac{2062038823585598216400886063744177152074382797460813233205628131}{4804049004628136761467451497915135586300439962085225710851838193}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{20}$, which has order $160$ (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:  \( 2417625.83378 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T40):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 16 conjugacy class representatives for $C_2\times C_2^4:C_5$
Character table for $C_2\times C_2^4:C_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.0.2670699013250048.1, 10.0.5048580365312.1, 10.10.116117348402176.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 sibling: data not computed
Degree 40 siblings: data not computed
Arithmetically equvalently 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 ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
$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$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$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$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$