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Properties

Label 20.0.32809060263...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 401^{10}$
Root discriminant $66.96$
Ramified primes $5, 401$
Class number $1616$ (GRH)
Class group $[2, 808]$ (GRH)
Galois group $D_{20}$ (as 20T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![282861, -1539672, 3798320, -5570593, 5546802, -4132047, 2500617, -1317875, 666734, -330321, 168816, -75285, 31833, -10617, 4645, -1163, 585, -108, 42, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 42*x^18 - 108*x^17 + 585*x^16 - 1163*x^15 + 4645*x^14 - 10617*x^13 + 31833*x^12 - 75285*x^11 + 168816*x^10 - 330321*x^9 + 666734*x^8 - 1317875*x^7 + 2500617*x^6 - 4132047*x^5 + 5546802*x^4 - 5570593*x^3 + 3798320*x^2 - 1539672*x + 282861)
 
gp: K = bnfinit(x^20 - 4*x^19 + 42*x^18 - 108*x^17 + 585*x^16 - 1163*x^15 + 4645*x^14 - 10617*x^13 + 31833*x^12 - 75285*x^11 + 168816*x^10 - 330321*x^9 + 666734*x^8 - 1317875*x^7 + 2500617*x^6 - 4132047*x^5 + 5546802*x^4 - 5570593*x^3 + 3798320*x^2 - 1539672*x + 282861, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 42 x^{18} - 108 x^{17} + 585 x^{16} - 1163 x^{15} + 4645 x^{14} - 10617 x^{13} + 31833 x^{12} - 75285 x^{11} + 168816 x^{10} - 330321 x^{9} + 666734 x^{8} - 1317875 x^{7} + 2500617 x^{6} - 4132047 x^{5} + 5546802 x^{4} - 5570593 x^{3} + 3798320 x^{2} - 1539672 x + 282861 \)

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:  \(3280906026328914297094848663330078125=5^{15}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.96$
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:  $5, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{2}{9} a^{6} - \frac{4}{9} a^{5} + \frac{2}{9} a^{4} - \frac{2}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{13} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{4}{9} a^{6} - \frac{2}{9} a^{5} - \frac{1}{9} a^{3} + \frac{4}{9} a^{2}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{2}{9} a^{6} - \frac{2}{9} a^{5} - \frac{1}{3} a^{4} - \frac{1}{9} a^{2}$, $\frac{1}{621} a^{17} + \frac{1}{207} a^{16} + \frac{8}{621} a^{15} + \frac{1}{23} a^{14} + \frac{100}{621} a^{13} - \frac{65}{621} a^{12} + \frac{103}{621} a^{11} + \frac{70}{621} a^{10} + \frac{103}{621} a^{9} - \frac{83}{621} a^{8} + \frac{43}{621} a^{7} + \frac{34}{621} a^{6} + \frac{11}{207} a^{5} + \frac{2}{27} a^{4} + \frac{305}{621} a^{3} + \frac{85}{621} a^{2} + \frac{83}{207} a + \frac{14}{69}$, $\frac{1}{169968321} a^{18} - \frac{35659}{56656107} a^{17} - \frac{7163593}{169968321} a^{16} + \frac{1452103}{56656107} a^{15} - \frac{5788499}{169968321} a^{14} - \frac{25547555}{169968321} a^{13} - \frac{22708262}{169968321} a^{12} + \frac{25093246}{169968321} a^{11} + \frac{339622}{169968321} a^{10} - \frac{21016640}{169968321} a^{9} + \frac{11496070}{169968321} a^{8} + \frac{13642942}{169968321} a^{7} + \frac{5233985}{18885369} a^{6} + \frac{18744838}{169968321} a^{5} - \frac{75006145}{169968321} a^{4} - \frac{19572014}{169968321} a^{3} - \frac{26257567}{56656107} a^{2} + \frac{106705}{320091} a + \frac{87352}{6295123}$, $\frac{1}{51106000946960374515929562567648605175177} a^{19} - \frac{84200404212130290377290862804357}{51106000946960374515929562567648605175177} a^{18} + \frac{1272304771364954075157534352783122852}{1892814849887421278367761576579577969451} a^{17} + \frac{2771805107874077692146616296527800021268}{51106000946960374515929562567648605175177} a^{16} - \frac{912891636975917774832980116635116096696}{17035333648986791505309854189216201725059} a^{15} - \frac{871491467867312166280381222037851574986}{51106000946960374515929562567648605175177} a^{14} + \frac{95596221114152067408228619910008237639}{17035333648986791505309854189216201725059} a^{13} + \frac{314219173777372935896491370383597602746}{1892814849887421278367761576579577969451} a^{12} - \frac{1186679143296649520565862151185435095214}{17035333648986791505309854189216201725059} a^{11} - \frac{108283061989917992448273333489900857014}{1310410280691291654254604168401246286543} a^{10} - \frac{794934066507936559441744530289209695108}{5678444549662263835103284729738733908353} a^{9} + \frac{2726769105805808935657324714933792119130}{17035333648986791505309854189216201725059} a^{8} - \frac{654301710385563830959206533327763477712}{3931230842073874962763812505203738859629} a^{7} + \frac{26598046841429748327477923370445971418}{74174166831582546467241745381202619993} a^{6} + \frac{1264405589483293676017359851566082657}{246888893463576688482751509988640604711} a^{5} - \frac{295286753822174074076769092910377626951}{2222000041172190196344763589897765442399} a^{4} - \frac{7163619266124056742871032818932340370502}{17035333648986791505309854189216201725059} a^{3} - \frac{12546639806499944656162198934176918667264}{51106000946960374515929562567648605175177} a^{2} + \frac{4673397936510930711841304122725194418347}{17035333648986791505309854189216201725059} a - \frac{42644245411103008825830961481873211398}{107140463201174789341571409995070451101}$

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

Class group and class number

$C_{2}\times C_{808}$, which has order $1616$ (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:  \( 31495162.1453 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{20}$ (as 20T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 13 conjugacy class representatives for $D_{20}$
Character table for $D_{20}$

Intermediate fields

\(\Q(\sqrt{2005}) \), 4.0.20100125.1, 5.5.160801.1, 10.10.32402005006253125.1

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 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 $20$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
5Data not computed
401Data not computed