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Properties

Label 20.0.59699157574...1344.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 11^{18}$
Root discriminant $24.48$
Ramified primes $2, 11$
Class number $2$
Class group $[2]$
Galois group $C_2^2\times C_2^4:C_5$ (as 20T74)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![529, -5980, 28094, -67450, 91781, -70730, 26778, -5132, 10032, -14490, 8590, -2446, 1331, -1674, 1048, -264, 7, -16, 22, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 22*x^18 - 16*x^17 + 7*x^16 - 264*x^15 + 1048*x^14 - 1674*x^13 + 1331*x^12 - 2446*x^11 + 8590*x^10 - 14490*x^9 + 10032*x^8 - 5132*x^7 + 26778*x^6 - 70730*x^5 + 91781*x^4 - 67450*x^3 + 28094*x^2 - 5980*x + 529)
 
gp: K = bnfinit(x^20 - 8*x^19 + 22*x^18 - 16*x^17 + 7*x^16 - 264*x^15 + 1048*x^14 - 1674*x^13 + 1331*x^12 - 2446*x^11 + 8590*x^10 - 14490*x^9 + 10032*x^8 - 5132*x^7 + 26778*x^6 - 70730*x^5 + 91781*x^4 - 67450*x^3 + 28094*x^2 - 5980*x + 529, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 22 x^{18} - 16 x^{17} + 7 x^{16} - 264 x^{15} + 1048 x^{14} - 1674 x^{13} + 1331 x^{12} - 2446 x^{11} + 8590 x^{10} - 14490 x^{9} + 10032 x^{8} - 5132 x^{7} + 26778 x^{6} - 70730 x^{5} + 91781 x^{4} - 67450 x^{3} + 28094 x^{2} - 5980 x + 529 \)

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:  \(5969915757478328440239161344=2^{30}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.48$
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:  $2, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3177264163715470389158282147} a^{19} - \frac{233359433245262983512987089}{3177264163715470389158282147} a^{18} - \frac{474790220912144945397895245}{3177264163715470389158282147} a^{17} + \frac{720789205931755020012080888}{3177264163715470389158282147} a^{16} - \frac{480034203956468188675436863}{3177264163715470389158282147} a^{15} + \frac{1455009824497072840985465546}{3177264163715470389158282147} a^{14} - \frac{1561117036708462137177184470}{3177264163715470389158282147} a^{13} + \frac{602601589199102281544903722}{3177264163715470389158282147} a^{12} + \frac{668948483468234794778432483}{3177264163715470389158282147} a^{11} + \frac{931228831183024112298131831}{3177264163715470389158282147} a^{10} - \frac{887064271235853923954261413}{3177264163715470389158282147} a^{9} + \frac{20800565887069360764475401}{138141920161542190832968789} a^{8} - \frac{35821784425254039591695358}{3177264163715470389158282147} a^{7} + \frac{1126722848908450234995136295}{3177264163715470389158282147} a^{6} - \frac{507289641603212982446303909}{3177264163715470389158282147} a^{5} - \frac{178389620243353485066607775}{3177264163715470389158282147} a^{4} - \frac{151592447455195165633450735}{3177264163715470389158282147} a^{3} + \frac{1052705059059540883667317238}{3177264163715470389158282147} a^{2} - \frac{457395286438301865408261820}{3177264163715470389158282147} a + \frac{5699978149499380496042384}{138141920161542190832968789}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 159636.716542 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_2^4:C_5$ (as 20T74):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_2^2\times C_2^4:C_5$
Character table for $C_2^2\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.2.2414538435584.1, 10.4.2414538435584.1, 10.4.219503494144.1

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
Arithmetically equvalently 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
11Data not computed