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Properties

Label 20.0.99425422805...6521.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 17^{14}$
Root discriminant $12.59$
Ramified primes $3, 17$
Class number $1$
Class group Trivial
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 16, -24, 69, -80, 149, -136, 188, -136, 149, -95, 100, -73, 74, -57, 47, -29, 15, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 15*x^18 - 29*x^17 + 47*x^16 - 57*x^15 + 74*x^14 - 73*x^13 + 100*x^12 - 95*x^11 + 149*x^10 - 136*x^9 + 188*x^8 - 136*x^7 + 149*x^6 - 80*x^5 + 69*x^4 - 24*x^3 + 16*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 15*x^18 - 29*x^17 + 47*x^16 - 57*x^15 + 74*x^14 - 73*x^13 + 100*x^12 - 95*x^11 + 149*x^10 - 136*x^9 + 188*x^8 - 136*x^7 + 149*x^6 - 80*x^5 + 69*x^4 - 24*x^3 + 16*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 15 x^{18} - 29 x^{17} + 47 x^{16} - 57 x^{15} + 74 x^{14} - 73 x^{13} + 100 x^{12} - 95 x^{11} + 149 x^{10} - 136 x^{9} + 188 x^{8} - 136 x^{7} + 149 x^{6} - 80 x^{5} + 69 x^{4} - 24 x^{3} + 16 x^{2} - 3 x + 1 \)

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:  \(9942542280506065456521=3^{10}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.59$
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:  $3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{16} - \frac{1}{3} a^{15} + \frac{1}{3} a^{14} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{38112433245} a^{19} - \frac{4186434373}{38112433245} a^{18} + \frac{10060526159}{38112433245} a^{17} - \frac{37838148}{184118035} a^{16} - \frac{1653555086}{7622486649} a^{15} - \frac{17354176822}{38112433245} a^{14} + \frac{63827092}{162180567} a^{13} - \frac{4006925491}{12704144415} a^{12} + \frac{4618575004}{38112433245} a^{11} + \frac{1005865913}{38112433245} a^{10} - \frac{3994759}{7622486649} a^{9} + \frac{3704178293}{12704144415} a^{8} + \frac{17912285531}{38112433245} a^{7} + \frac{6855274291}{38112433245} a^{6} - \frac{5161566428}{12704144415} a^{5} + \frac{104574536}{810902835} a^{4} + \frac{7428927793}{38112433245} a^{3} - \frac{15439978508}{38112433245} a^{2} - \frac{80782096}{162180567} a - \frac{12735182278}{38112433245}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{2199359}{6826515} a^{19} - \frac{12662477}{6826515} a^{18} + \frac{37433461}{6826515} a^{17} - \frac{1001541}{98935} a^{16} + \frac{18836408}{1365303} a^{15} - \frac{96521393}{6826515} a^{14} + \frac{401297}{29049} a^{13} - \frac{33027459}{2275505} a^{12} + \frac{116561426}{6826515} a^{11} - \frac{159631478}{6826515} a^{10} + \frac{38189908}{1365303} a^{9} - \frac{83580553}{2275505} a^{8} + \frac{187307989}{6826515} a^{7} - \frac{175133461}{6826515} a^{6} + \frac{6857713}{2275505} a^{5} - \frac{646031}{145245} a^{4} - \frac{49555663}{6826515} a^{3} + \frac{12422858}{6826515} a^{2} - \frac{107555}{29049} a + \frac{5908858}{6826515} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1272.4712763 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{-3}, \sqrt{17})\), 5.1.44217.1, 10.0.5865429267.1, 10.2.33237432513.1, 10.0.99712297539.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$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$$[\ ]_{2}^{4}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$