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Properties

Label 20.0.12203667882...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{22}\cdot 13^{15}$
Root discriminant $40.21$
Ramified primes $5, 13$
Class number $2$ (GRH)
Class group $[2]$ (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![426416, -1907160, 3414000, -3571940, 3162985, -2758493, 2000995, -1199615, 694400, -352875, 143098, -53335, 16815, -3420, 615, -158, 75, -75, 40, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 40*x^18 - 75*x^17 + 75*x^16 - 158*x^15 + 615*x^14 - 3420*x^13 + 16815*x^12 - 53335*x^11 + 143098*x^10 - 352875*x^9 + 694400*x^8 - 1199615*x^7 + 2000995*x^6 - 2758493*x^5 + 3162985*x^4 - 3571940*x^3 + 3414000*x^2 - 1907160*x + 426416)
 
gp: K = bnfinit(x^20 - 10*x^19 + 40*x^18 - 75*x^17 + 75*x^16 - 158*x^15 + 615*x^14 - 3420*x^13 + 16815*x^12 - 53335*x^11 + 143098*x^10 - 352875*x^9 + 694400*x^8 - 1199615*x^7 + 2000995*x^6 - 2758493*x^5 + 3162985*x^4 - 3571940*x^3 + 3414000*x^2 - 1907160*x + 426416, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 40 x^{18} - 75 x^{17} + 75 x^{16} - 158 x^{15} + 615 x^{14} - 3420 x^{13} + 16815 x^{12} - 53335 x^{11} + 143098 x^{10} - 352875 x^{9} + 694400 x^{8} - 1199615 x^{7} + 2000995 x^{6} - 2758493 x^{5} + 3162985 x^{4} - 3571940 x^{3} + 3414000 x^{2} - 1907160 x + 426416 \)

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:  \(122036678824641125202178955078125=5^{22}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.21$
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:  $5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{16} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} + \frac{1}{8} a^{13} + \frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{4} a^{9} - \frac{3}{8} a^{8} + \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{16} - \frac{1}{4} a^{15} - \frac{1}{8} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} + \frac{1}{8} a^{11} - \frac{1}{8} a^{9} + \frac{1}{4} a^{8} + \frac{1}{8} a^{6} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{18198701516734855765093550260450294132480575688628456} a^{19} + \frac{272024871174578269463746247118147949160387919158417}{4549675379183713941273387565112573533120143922157114} a^{18} - \frac{123728448294857984979883079267616431557140949186587}{2274837689591856970636693782556286766560071961078557} a^{17} + \frac{471374844840989020099446090845386276491828545477475}{18198701516734855765093550260450294132480575688628456} a^{16} + \frac{2269743079723664173173791641547771737795673069322819}{18198701516734855765093550260450294132480575688628456} a^{15} - \frac{955556218996186509364361179658840118681597466532033}{4549675379183713941273387565112573533120143922157114} a^{14} - \frac{3774751748654836796524531176965593205795657344401793}{18198701516734855765093550260450294132480575688628456} a^{13} - \frac{29134992973401221195099705306846316685158256087020}{2274837689591856970636693782556286766560071961078557} a^{12} - \frac{1359925402866896651497805958559708286238810430318519}{18198701516734855765093550260450294132480575688628456} a^{11} + \frac{8380336579592545024219369722357347774430774111266141}{18198701516734855765093550260450294132480575688628456} a^{10} - \frac{1412317523036635077685232501981619443126184442670205}{4549675379183713941273387565112573533120143922157114} a^{9} - \frac{1986491321178486383377862376136550219891173824763251}{18198701516734855765093550260450294132480575688628456} a^{8} + \frac{588960985127574298070346023088328346049312237050020}{2274837689591856970636693782556286766560071961078557} a^{7} - \frac{3036523525188674734386884544500121885202469545916741}{18198701516734855765093550260450294132480575688628456} a^{6} + \frac{19228893450871794634564808094737705667689529485755}{105194806455114773208633238499712682846708529992072} a^{5} - \frac{8442073519895215533611403357296958410940789854512939}{18198701516734855765093550260450294132480575688628456} a^{4} + \frac{953828432147308040816186811733186304781840816967979}{18198701516734855765093550260450294132480575688628456} a^{3} + \frac{559416611878779697489176202557526846848202724095579}{4549675379183713941273387565112573533120143922157114} a^{2} - \frac{608673368287449317449794399989934040060519905414276}{2274837689591856970636693782556286766560071961078557} a - \frac{203066953596380034689350187087307646609737314809746}{2274837689591856970636693782556286766560071961078557}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 54862279.394896545 \) (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

\(\Q(\sqrt{13}) \), 4.0.54925.1, 5.1.528125.1, 10.2.3625908203125.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
13Data not computed