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Properties

Label 20.0.18764086632...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{15}\cdot 19^{18}$
Root discriminant $81.97$
Ramified primes $3, 5, 19$
Class number $32$ (GRH)
Class group $[2, 2, 2, 2, 2]$ (GRH)
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![8509591, 24180423, 40765960, 54483025, 51034338, 35686946, 17183403, 4440981, 1796869, 433083, -45091, -57053, -25967, -8469, 1705, -440, -294, 97, 26, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 26*x^18 + 97*x^17 - 294*x^16 - 440*x^15 + 1705*x^14 - 8469*x^13 - 25967*x^12 - 57053*x^11 - 45091*x^10 + 433083*x^9 + 1796869*x^8 + 4440981*x^7 + 17183403*x^6 + 35686946*x^5 + 51034338*x^4 + 54483025*x^3 + 40765960*x^2 + 24180423*x + 8509591)
 
gp: K = bnfinit(x^20 - 3*x^19 + 26*x^18 + 97*x^17 - 294*x^16 - 440*x^15 + 1705*x^14 - 8469*x^13 - 25967*x^12 - 57053*x^11 - 45091*x^10 + 433083*x^9 + 1796869*x^8 + 4440981*x^7 + 17183403*x^6 + 35686946*x^5 + 51034338*x^4 + 54483025*x^3 + 40765960*x^2 + 24180423*x + 8509591, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 26 x^{18} + 97 x^{17} - 294 x^{16} - 440 x^{15} + 1705 x^{14} - 8469 x^{13} - 25967 x^{12} - 57053 x^{11} - 45091 x^{10} + 433083 x^{9} + 1796869 x^{8} + 4440981 x^{7} + 17183403 x^{6} + 35686946 x^{5} + 51034338 x^{4} + 54483025 x^{3} + 40765960 x^{2} + 24180423 x + 8509591 \)

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:  \(187640866325114773598410895050048828125=3^{10}\cdot 5^{15}\cdot 19^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.97$
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:  $3, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{82} a^{15} - \frac{3}{41} a^{14} + \frac{5}{41} a^{13} - \frac{2}{41} a^{12} + \frac{14}{41} a^{11} + \frac{5}{41} a^{10} + \frac{19}{41} a^{9} - \frac{10}{41} a^{8} - \frac{3}{41} a^{7} - \frac{2}{41} a^{5} - \frac{20}{41} a^{4} + \frac{18}{41} a^{3} + \frac{7}{41} a^{2} - \frac{13}{41} a - \frac{1}{2}$, $\frac{1}{410} a^{16} + \frac{97}{410} a^{14} + \frac{3}{82} a^{13} + \frac{127}{410} a^{12} + \frac{11}{82} a^{11} - \frac{5}{82} a^{10} + \frac{17}{82} a^{9} + \frac{161}{410} a^{8} + \frac{1}{82} a^{7} - \frac{9}{82} a^{6} - \frac{21}{82} a^{5} + \frac{83}{410} a^{4} + \frac{5}{82} a^{3} + \frac{181}{410} a^{2} + \frac{9}{41} a + \frac{1}{10}$, $\frac{1}{410} a^{17} + \frac{1}{205} a^{15} - \frac{3}{41} a^{14} + \frac{101}{205} a^{13} - \frac{18}{41} a^{12} - \frac{2}{41} a^{11} + \frac{16}{41} a^{10} + \frac{18}{205} a^{9} + \frac{6}{41} a^{8} - \frac{9}{41} a^{7} + \frac{10}{41} a^{6} - \frac{76}{205} a^{5} - \frac{7}{41} a^{4} - \frac{2}{5} a^{3} + \frac{39}{82} a^{2} - \frac{77}{205} a$, $\frac{1}{410} a^{18} + \frac{33}{410} a^{14} - \frac{23}{82} a^{13} - \frac{189}{410} a^{12} - \frac{27}{82} a^{11} + \frac{181}{410} a^{10} + \frac{1}{82} a^{9} + \frac{13}{410} a^{8} + \frac{23}{82} a^{7} + \frac{143}{410} a^{6} - \frac{37}{82} a^{5} - \frac{19}{82} a^{4} + \frac{20}{41} a^{3} + \frac{109}{410} a^{2} + \frac{13}{82} a + \frac{3}{10}$, $\frac{1}{1577938019948434506029426619293093585364523757926953820065763085560450} a^{19} - \frac{211817111778911475506783901461218790484446318498526240804881136461}{1577938019948434506029426619293093585364523757926953820065763085560450} a^{18} - \frac{1045443455497261369927841522247167051387773602754712374019344299451}{1577938019948434506029426619293093585364523757926953820065763085560450} a^{17} - \frac{93511841533650751840645591751822625014397200570618576305592152411}{157793801994843450602942661929309358536452375792695382006576308556045} a^{16} - \frac{36338466679769991319533315222280743376870252889106760453912592437}{788969009974217253014713309646546792682261878963476910032881542780225} a^{15} - \frac{140683582577529383809629710881138471180649765048963295149852410928453}{1577938019948434506029426619293093585364523757926953820065763085560450} a^{14} - \frac{440666558047418773379728643456214300049100917748781458312787914066911}{1577938019948434506029426619293093585364523757926953820065763085560450} a^{13} - \frac{271794940555384348844593414197606212846066490771979005403525436055861}{1577938019948434506029426619293093585364523757926953820065763085560450} a^{12} + \frac{25210378511438450837157723975757540313864874077954441212049139199811}{1577938019948434506029426619293093585364523757926953820065763085560450} a^{11} + \frac{314728806536723939105572011459046741766834315470134259816364261705109}{1577938019948434506029426619293093585364523757926953820065763085560450} a^{10} + \frac{280022039704079906499826949829033302744297330524910469705058995222297}{1577938019948434506029426619293093585364523757926953820065763085560450} a^{9} - \frac{10489387050592635449700700490735525993476525583313207348634660399433}{1577938019948434506029426619293093585364523757926953820065763085560450} a^{8} + \frac{603212395458662747800817282414484963057085141308692776795584046216953}{1577938019948434506029426619293093585364523757926953820065763085560450} a^{7} - \frac{373170133017895727829811853259353370939281920586602664445936957440143}{1577938019948434506029426619293093585364523757926953820065763085560450} a^{6} + \frac{504557344488332746885249689685632926115222521878916217588483781920177}{1577938019948434506029426619293093585364523757926953820065763085560450} a^{5} - \frac{30132073693548534340961703822049500901384460481350811772419785620264}{157793801994843450602942661929309358536452375792695382006576308556045} a^{4} - \frac{99194975497188681837561632187734887462885301934612240743848630203186}{788969009974217253014713309646546792682261878963476910032881542780225} a^{3} + \frac{36054615038305185001495857380253455294115386437203841297364373332068}{788969009974217253014713309646546792682261878963476910032881542780225} a^{2} + \frac{710000846981202553931442304739177712209755592008359767726755215790777}{1577938019948434506029426619293093585364523757926953820065763085560450} a + \frac{1053125201908357907523837292210222220352089334054614566681661639137}{38486293169474012342181137055929111838159116046998873660140563062450}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $32$ (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:  \( 3932635547.52714 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), 4.0.406125.2, 5.1.16290125.1, 10.2.1326840862578125.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$19.10.9.1$x^{10} - 19$$10$$1$$9$$D_{10}$$[\ ]_{10}^{2}$
19.10.9.1$x^{10} - 19$$10$$1$$9$$D_{10}$$[\ ]_{10}^{2}$