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Properties

Label 20.0.22391715889...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{31}\cdot 37^{10}$
Root discriminant $73.71$
Ramified primes $5, 37$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![401120497321, 490733138390, 322173998635, 130244452175, 26632021340, -4302198659, -5269012810, -2228991105, -452982025, -34682135, 23745326, 8083750, 2145955, 174325, 33025, -3619, 1370, -110, 50, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 50*x^18 - 110*x^17 + 1370*x^16 - 3619*x^15 + 33025*x^14 + 174325*x^13 + 2145955*x^12 + 8083750*x^11 + 23745326*x^10 - 34682135*x^9 - 452982025*x^8 - 2228991105*x^7 - 5269012810*x^6 - 4302198659*x^5 + 26632021340*x^4 + 130244452175*x^3 + 322173998635*x^2 + 490733138390*x + 401120497321)
 
gp: K = bnfinit(x^20 - 5*x^19 + 50*x^18 - 110*x^17 + 1370*x^16 - 3619*x^15 + 33025*x^14 + 174325*x^13 + 2145955*x^12 + 8083750*x^11 + 23745326*x^10 - 34682135*x^9 - 452982025*x^8 - 2228991105*x^7 - 5269012810*x^6 - 4302198659*x^5 + 26632021340*x^4 + 130244452175*x^3 + 322173998635*x^2 + 490733138390*x + 401120497321, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 50 x^{18} - 110 x^{17} + 1370 x^{16} - 3619 x^{15} + 33025 x^{14} + 174325 x^{13} + 2145955 x^{12} + 8083750 x^{11} + 23745326 x^{10} - 34682135 x^{9} - 452982025 x^{8} - 2228991105 x^{7} - 5269012810 x^{6} - 4302198659 x^{5} + 26632021340 x^{4} + 130244452175 x^{3} + 322173998635 x^{2} + 490733138390 x + 401120497321 \)

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:  \(22391715889879730530083179473876953125=5^{31}\cdot 37^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.71$
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:  $5, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{76} a^{17} + \frac{5}{76} a^{16} + \frac{15}{76} a^{15} - \frac{1}{4} a^{14} + \frac{3}{38} a^{13} - \frac{7}{76} a^{12} + \frac{3}{19} a^{11} - \frac{15}{76} a^{10} - \frac{27}{76} a^{9} + \frac{15}{38} a^{8} - \frac{7}{76} a^{7} - \frac{5}{38} a^{6} - \frac{5}{76} a^{5} - \frac{1}{76} a^{4} - \frac{7}{19} a^{3} + \frac{5}{19} a^{2} + \frac{23}{76} a - \frac{3}{19}$, $\frac{1}{152} a^{18} + \frac{9}{152} a^{16} - \frac{37}{152} a^{15} - \frac{4}{19} a^{14} - \frac{9}{76} a^{13} - \frac{29}{152} a^{12} - \frac{9}{76} a^{11} + \frac{5}{76} a^{10} + \frac{35}{76} a^{9} - \frac{31}{76} a^{8} + \frac{25}{152} a^{7} + \frac{13}{76} a^{6} - \frac{13}{38} a^{5} - \frac{1}{38} a^{4} - \frac{11}{152} a^{3} + \frac{37}{152} a^{2} - \frac{13}{152} a - \frac{35}{152}$, $\frac{1}{25650229661609849845098870045485565820033600536372845701464680235879371188728327725917031894456560552} a^{19} - \frac{14631875886277043108342302656670615389644504068813892129337098174576201569298024898398100203440207}{12825114830804924922549435022742782910016800268186422850732340117939685594364163862958515947228280276} a^{18} - \frac{58421832954286931337383004924039509884128780099233934843406398816167467452386334549289605364592685}{25650229661609849845098870045485565820033600536372845701464680235879371188728327725917031894456560552} a^{17} - \frac{214798694912641685067298545218885150151981724176192901396389582890324581977179744218479065130058509}{25650229661609849845098870045485565820033600536372845701464680235879371188728327725917031894456560552} a^{16} - \frac{763479524571345791833013761033238136060729328722320804860999633431493124562754251269859272649301267}{3206278707701231230637358755685695727504200067046605712683085029484921398591040965739628986807070069} a^{15} - \frac{579187868331790060133891740061340894389759809348580736894825244079930268038216451435102337560025738}{3206278707701231230637358755685695727504200067046605712683085029484921398591040965739628986807070069} a^{14} - \frac{6314019424488051776190330190111390533472726721429372900818895543327685874027241688732676894675001901}{25650229661609849845098870045485565820033600536372845701464680235879371188728327725917031894456560552} a^{13} + \frac{1303307404578155264197176171914754416145970424970119743035865291935835726152990009425906546458764491}{12825114830804924922549435022742782910016800268186422850732340117939685594364163862958515947228280276} a^{12} - \frac{860577145347341906771951259077767202837910995481060312482681571554187324282664607293750649474077777}{12825114830804924922549435022742782910016800268186422850732340117939685594364163862958515947228280276} a^{11} + \frac{347809604048250558164132540265721320842360738391263783843542080363542466111993078145342540626017876}{3206278707701231230637358755685695727504200067046605712683085029484921398591040965739628986807070069} a^{10} - \frac{386231389888188641856837721188158626367666601797278151442886043622803520172888475329714899395774792}{3206278707701231230637358755685695727504200067046605712683085029484921398591040965739628986807070069} a^{9} + \frac{305238511394184407335545333747124925249924317454075289729685999742363318239464160613496121313551917}{884490677986546546382719656740881580001158639185270541429816559857909351335459576755759720498502088} a^{8} - \frac{4755515654713893028335698135729996214166498244158692958858001750882722542213375073165423154903449021}{12825114830804924922549435022742782910016800268186422850732340117939685594364163862958515947228280276} a^{7} + \frac{588671422614771658459055402783471891889331086732558514517502261856314612274900386300121546151480309}{6412557415402462461274717511371391455008400134093211425366170058969842797182081931479257973614140138} a^{6} - \frac{2865683706103797466630630543057264885786207744384354618419496640449209725311025891049995329198012869}{12825114830804924922549435022742782910016800268186422850732340117939685594364163862958515947228280276} a^{5} + \frac{4126212062908919649982650840494940033360835125207116084035684999082355566546203515979616695874805059}{25650229661609849845098870045485565820033600536372845701464680235879371188728327725917031894456560552} a^{4} + \frac{8833642650722133806817290501640734895119735669336542315450843834551933570297667839754722144655690151}{25650229661609849845098870045485565820033600536372845701464680235879371188728327725917031894456560552} a^{3} - \frac{5158599992548244408797174549905024983156723279523084792304626596970437806855346110177325000562079299}{25650229661609849845098870045485565820033600536372845701464680235879371188728327725917031894456560552} a^{2} + \frac{10700871642350665364588686723243965267617378754240513088476237334533291181852247645797201702755142485}{25650229661609849845098870045485565820033600536372845701464680235879371188728327725917031894456560552} a - \frac{4219163428273382059224859517608525558721766364287580277459385391871400126114275913711061824873847187}{12825114830804924922549435022742782910016800268186422850732340117939685594364163862958515947228280276}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.171125.1, 5.1.78125.1, 10.2.30517578125.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
$37$37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$