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Properties

Label 20.12.5331335356...0625.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{10}\cdot 11^{16}\cdot 109^{2}$
Root discriminant $24.34$
Ramified primes $5, 11, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2\times C_2^4:C_5$ (as 20T86)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -19, 14, 65, -76, -102, 79, 198, 261, -158, -648, -363, 125, 167, 21, -4, 7, -6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 6*x^18 + 7*x^17 - 4*x^16 + 21*x^15 + 167*x^14 + 125*x^13 - 363*x^12 - 648*x^11 - 158*x^10 + 261*x^9 + 198*x^8 + 79*x^7 - 102*x^6 - 76*x^5 + 65*x^4 + 14*x^3 - 19*x^2 + x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 - 6*x^18 + 7*x^17 - 4*x^16 + 21*x^15 + 167*x^14 + 125*x^13 - 363*x^12 - 648*x^11 - 158*x^10 + 261*x^9 + 198*x^8 + 79*x^7 - 102*x^6 - 76*x^5 + 65*x^4 + 14*x^3 - 19*x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 6 x^{18} + 7 x^{17} - 4 x^{16} + 21 x^{15} + 167 x^{14} + 125 x^{13} - 363 x^{12} - 648 x^{11} - 158 x^{10} + 261 x^{9} + 198 x^{8} + 79 x^{7} - 102 x^{6} - 76 x^{5} + 65 x^{4} + 14 x^{3} - 19 x^{2} + x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5331335356534187937900390625=5^{10}\cdot 11^{16}\cdot 109^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.34$
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:  $5, 11, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{89} a^{17} - \frac{18}{89} a^{16} - \frac{11}{89} a^{15} + \frac{29}{89} a^{14} + \frac{9}{89} a^{13} - \frac{37}{89} a^{12} - \frac{19}{89} a^{11} - \frac{8}{89} a^{10} + \frac{26}{89} a^{9} + \frac{29}{89} a^{8} - \frac{18}{89} a^{7} - \frac{43}{89} a^{6} - \frac{38}{89} a^{5} + \frac{18}{89} a^{4} - \frac{9}{89} a^{3} - \frac{37}{89} a^{2} - \frac{24}{89} a - \frac{40}{89}$, $\frac{1}{89} a^{18} + \frac{21}{89} a^{16} + \frac{9}{89} a^{15} - \frac{3}{89} a^{14} + \frac{36}{89} a^{13} + \frac{27}{89} a^{12} + \frac{6}{89} a^{11} - \frac{29}{89} a^{10} - \frac{37}{89} a^{9} - \frac{30}{89} a^{8} - \frac{11}{89} a^{7} - \frac{11}{89} a^{6} - \frac{43}{89} a^{5} - \frac{41}{89} a^{4} - \frac{21}{89} a^{3} + \frac{22}{89} a^{2} - \frac{27}{89} a - \frac{8}{89}$, $\frac{1}{1826782984772789} a^{19} + \frac{9888204408796}{1826782984772789} a^{18} + \frac{8598613758930}{1826782984772789} a^{17} - \frac{305186805813206}{1826782984772789} a^{16} + \frac{499323336843997}{1826782984772789} a^{15} + \frac{16956039319233}{1826782984772789} a^{14} + \frac{255448154670436}{1826782984772789} a^{13} - \frac{175501539549135}{1826782984772789} a^{12} + \frac{881417674900947}{1826782984772789} a^{11} - \frac{130077990431168}{1826782984772789} a^{10} + \frac{280023991924044}{1826782984772789} a^{9} + \frac{88417845660029}{1826782984772789} a^{8} + \frac{2034966338711}{16759476924521} a^{7} - \frac{53869195296604}{1826782984772789} a^{6} - \frac{567088938614130}{1826782984772789} a^{5} - \frac{276840932552154}{1826782984772789} a^{4} - \frac{507768904477272}{1826782984772789} a^{3} - \frac{750308682924895}{1826782984772789} a^{2} - \frac{235016392227541}{1826782984772789} a - \frac{562328666234150}{1826782984772789}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 3491062.30074 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_2^4:C_5$ (as 20T86):

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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{11})^+\), 10.6.73015993840625.1, 10.6.23365118029.1, 10.10.669871503125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$109$$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$