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Properties

Label 20.0.76237898330...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 61^{7}\cdot 397^{4}$
Root discriminant $31.20$
Ramified primes $5, 61, 397$
Class number $16$ (GRH)
Class group $[2, 8]$ (GRH)
Galois group 20T174

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![139, 900, 2598, 4568, 6763, 8419, 9309, 7447, 6133, 2965, 2595, 812, 1028, -45, 169, 26, 29, -9, 11, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 11*x^18 - 9*x^17 + 29*x^16 + 26*x^15 + 169*x^14 - 45*x^13 + 1028*x^12 + 812*x^11 + 2595*x^10 + 2965*x^9 + 6133*x^8 + 7447*x^7 + 9309*x^6 + 8419*x^5 + 6763*x^4 + 4568*x^3 + 2598*x^2 + 900*x + 139)
 
gp: K = bnfinit(x^20 - 3*x^19 + 11*x^18 - 9*x^17 + 29*x^16 + 26*x^15 + 169*x^14 - 45*x^13 + 1028*x^12 + 812*x^11 + 2595*x^10 + 2965*x^9 + 6133*x^8 + 7447*x^7 + 9309*x^6 + 8419*x^5 + 6763*x^4 + 4568*x^3 + 2598*x^2 + 900*x + 139, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 11 x^{18} - 9 x^{17} + 29 x^{16} + 26 x^{15} + 169 x^{14} - 45 x^{13} + 1028 x^{12} + 812 x^{11} + 2595 x^{10} + 2965 x^{9} + 6133 x^{8} + 7447 x^{7} + 9309 x^{6} + 8419 x^{5} + 6763 x^{4} + 4568 x^{3} + 2598 x^{2} + 900 x + 139 \)

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:  \(762378983303206514165048828125=5^{10}\cdot 61^{7}\cdot 397^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.20$
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:  $5, 61, 397$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{1937} a^{18} - \frac{632}{1937} a^{17} + \frac{280}{1937} a^{16} - \frac{303}{1937} a^{15} + \frac{495}{1937} a^{14} + \frac{951}{1937} a^{13} - \frac{379}{1937} a^{12} - \frac{734}{1937} a^{11} - \frac{141}{1937} a^{10} + \frac{21}{149} a^{9} + \frac{687}{1937} a^{8} - \frac{158}{1937} a^{7} - \frac{464}{1937} a^{6} - \frac{558}{1937} a^{5} - \frac{609}{1937} a^{4} + \frac{448}{1937} a^{3} - \frac{544}{1937} a^{2} - \frac{451}{1937} a - \frac{656}{1937}$, $\frac{1}{9183015676650977259311449446135497} a^{19} + \frac{607487333343735335859331452}{46614292774878057153865225614901} a^{18} + \frac{4451981327645068940149618672092336}{9183015676650977259311449446135497} a^{17} + \frac{841630269915031988456851429509991}{9183015676650977259311449446135497} a^{16} - \frac{3642798778270794781154585842679820}{9183015676650977259311449446135497} a^{15} - \frac{163504781129441779076949554022761}{706385821280844404562419188164269} a^{14} + \frac{3835769940572642208983627828470690}{9183015676650977259311449446135497} a^{13} - \frac{4307579803563337096894197623213367}{9183015676650977259311449446135497} a^{12} + \frac{3395469257127183181396256938903174}{9183015676650977259311449446135497} a^{11} + \frac{1012263138044912858611480031565555}{9183015676650977259311449446135497} a^{10} - \frac{286350221073596907595654386417865}{9183015676650977259311449446135497} a^{9} - \frac{3179791726343094441992054996685615}{9183015676650977259311449446135497} a^{8} - \frac{3268414112410404747452242300305330}{9183015676650977259311449446135497} a^{7} + \frac{2582165871151240595901422672652664}{9183015676650977259311449446135497} a^{6} - \frac{17997117510284507489052968606908}{64216892843713127687492653469479} a^{5} - \frac{2004978633113812474145303179275786}{9183015676650977259311449446135497} a^{4} - \frac{3109889997849886722035507327931699}{9183015676650977259311449446135497} a^{3} + \frac{269718973816544108530611256742649}{706385821280844404562419188164269} a^{2} - \frac{128700866158326696873474987602470}{834819606968270659937404495103227} a + \frac{3441195245601199936353446714644794}{9183015676650977259311449446135497}$

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

Class group and class number

$C_{2}\times C_{8}$, which has order $16$ (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:  \( 266746.74584 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T174:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 960
The 35 conjugacy class representatives for t20n174
Character table for t20n174 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.1525.1, 5.5.24217.1, 10.10.1832697153125.1

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 24 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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
397Data not computed