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Properties

Label 20.0.37603126978...8125.2
Degree $20$
Signature $[0, 10]$
Discriminant $3^{16}\cdot 5^{15}\cdot 17^{15}$
Root discriminant $67.42$
Ramified primes $3, 5, 17$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65098640, -28170920, -19502880, 26088780, -10987249, 4125524, 1460216, -1893958, 1189621, -628428, 260740, -65765, 23172, -430, -925, 896, -304, 49, 4, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 4*x^18 + 49*x^17 - 304*x^16 + 896*x^15 - 925*x^14 - 430*x^13 + 23172*x^12 - 65765*x^11 + 260740*x^10 - 628428*x^9 + 1189621*x^8 - 1893958*x^7 + 1460216*x^6 + 4125524*x^5 - 10987249*x^4 + 26088780*x^3 - 19502880*x^2 - 28170920*x + 65098640)
 
gp: K = bnfinit(x^20 - 3*x^19 + 4*x^18 + 49*x^17 - 304*x^16 + 896*x^15 - 925*x^14 - 430*x^13 + 23172*x^12 - 65765*x^11 + 260740*x^10 - 628428*x^9 + 1189621*x^8 - 1893958*x^7 + 1460216*x^6 + 4125524*x^5 - 10987249*x^4 + 26088780*x^3 - 19502880*x^2 - 28170920*x + 65098640, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 4 x^{18} + 49 x^{17} - 304 x^{16} + 896 x^{15} - 925 x^{14} - 430 x^{13} + 23172 x^{12} - 65765 x^{11} + 260740 x^{10} - 628428 x^{9} + 1189621 x^{8} - 1893958 x^{7} + 1460216 x^{6} + 4125524 x^{5} - 10987249 x^{4} + 26088780 x^{3} - 19502880 x^{2} - 28170920 x + 65098640 \)

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:  \(3760312697824452795491478057861328125=3^{16}\cdot 5^{15}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.42$
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:  $3, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{15} + \frac{1}{6} a^{14} + \frac{1}{6} a^{13} - \frac{1}{2} a^{12} + \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{2} a^{9} + \frac{1}{6} a^{8} - \frac{1}{2} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{6} a^{16} - \frac{1}{6} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{6} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{17} - \frac{1}{6} a^{14} + \frac{1}{6} a^{13} - \frac{1}{3} a^{12} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} + \frac{1}{3} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a$, $\frac{1}{12} a^{18} - \frac{1}{12} a^{17} - \frac{1}{12} a^{15} + \frac{1}{6} a^{14} - \frac{1}{4} a^{12} + \frac{1}{3} a^{10} + \frac{5}{12} a^{9} - \frac{1}{2} a^{8} - \frac{1}{3} a^{7} + \frac{1}{4} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{3} - \frac{1}{12} a^{2} - \frac{1}{3} a$, $\frac{1}{546645493224105492662392874001618824391391984208474128978475480333628504} a^{19} - \frac{14248653946929169796182759962989303843629069671123230290360558273278743}{546645493224105492662392874001618824391391984208474128978475480333628504} a^{18} + \frac{10603612331580392196993575463584126498432031037892081181930959211992225}{136661373306026373165598218500404706097847996052118532244618870083407126} a^{17} - \frac{19297844378723840784738366317286270170306032265257652976509098037872135}{546645493224105492662392874001618824391391984208474128978475480333628504} a^{16} - \frac{2100448184997456531396419604217731945420368012442086315896697197770871}{45553791102008791055199406166801568699282665350706177414872956694469042} a^{15} - \frac{1973485370854872905083395788424621353434969970771180871691484069624322}{22776895551004395527599703083400784349641332675353088707436478347234521} a^{14} - \frac{75349527846499771946606291071562344313827485598740108951123873598895081}{546645493224105492662392874001618824391391984208474128978475480333628504} a^{13} + \frac{83441499564851091152494426795196561232701455586994518238928965107946671}{273322746612052746331196437000809412195695992104237064489237740166814252} a^{12} + \frac{2634649483402624264053684933932399915789352577679882998587843381723365}{136661373306026373165598218500404706097847996052118532244618870083407126} a^{11} + \frac{214454451540887392005527737624607107485445433003733075083335842844715519}{546645493224105492662392874001618824391391984208474128978475480333628504} a^{10} + \frac{7247087335350399552230574843911751833043528797427076092756103680994508}{22776895551004395527599703083400784349641332675353088707436478347234521} a^{9} - \frac{5928968122620766129652615788918131324842609673079265574072037220959259}{45553791102008791055199406166801568699282665350706177414872956694469042} a^{8} - \frac{219561402809970182390046390352333208075217787748387097140211024010001383}{546645493224105492662392874001618824391391984208474128978475480333628504} a^{7} - \frac{3799438518384148416487577425861852501789170010027665862597084080386969}{273322746612052746331196437000809412195695992104237064489237740166814252} a^{6} + \frac{2416942214468895176818270131269885224997818897104712827832941441289271}{68330686653013186582799109250202353048923998026059266122309435041703563} a^{5} - \frac{17817001285335138238636786994886898689266695948735479246216533645333285}{68330686653013186582799109250202353048923998026059266122309435041703563} a^{4} + \frac{173752919241455570902473128451905329840282980228840562118639438680617535}{546645493224105492662392874001618824391391984208474128978475480333628504} a^{3} + \frac{52795510824407056846334733919258700867958003669828750140744837150925227}{136661373306026373165598218500404706097847996052118532244618870083407126} a^{2} - \frac{62986001120551757078067855864120117538668872880677103820587340846942615}{136661373306026373165598218500404706097847996052118532244618870083407126} a + \frac{305655546632288410416228955072194389744464133752972928853336825911949}{22776895551004395527599703083400784349641332675353088707436478347234521}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 5918089885.052663 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{85}) \), 4.0.614125.2, 5.1.2926125.1, 10.2.727787638828125.1

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

Sibling fields

Degree 20 siblings: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
5Data not computed
17Data not computed