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Properties

Label 20.4.13483238599...4976.7
Degree $20$
Signature $[4, 8]$
Discriminant $2^{20}\cdot 11^{16}\cdot 23^{4}$
Root discriminant $25.50$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T751

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7613, 25944, -38321, 25224, 8013, -35210, 37486, -18940, -2698, 12802, -9977, 2626, 1688, -1854, 580, 136, -176, 48, 6, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 6*x^18 + 48*x^17 - 176*x^16 + 136*x^15 + 580*x^14 - 1854*x^13 + 1688*x^12 + 2626*x^11 - 9977*x^10 + 12802*x^9 - 2698*x^8 - 18940*x^7 + 37486*x^6 - 35210*x^5 + 8013*x^4 + 25224*x^3 - 38321*x^2 + 25944*x - 7613)
 
gp: K = bnfinit(x^20 - 6*x^19 + 6*x^18 + 48*x^17 - 176*x^16 + 136*x^15 + 580*x^14 - 1854*x^13 + 1688*x^12 + 2626*x^11 - 9977*x^10 + 12802*x^9 - 2698*x^8 - 18940*x^7 + 37486*x^6 - 35210*x^5 + 8013*x^4 + 25224*x^3 - 38321*x^2 + 25944*x - 7613, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 6 x^{18} + 48 x^{17} - 176 x^{16} + 136 x^{15} + 580 x^{14} - 1854 x^{13} + 1688 x^{12} + 2626 x^{11} - 9977 x^{10} + 12802 x^{9} - 2698 x^{8} - 18940 x^{7} + 37486 x^{6} - 35210 x^{5} + 8013 x^{4} + 25224 x^{3} - 38321 x^{2} + 25944 x - 7613 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13483238599952325260241534976=2^{20}\cdot 11^{16}\cdot 23^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.50$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1937241596269246950012279629010020801} a^{19} + \frac{97192399413813818992256857016719109}{1937241596269246950012279629010020801} a^{18} + \frac{893259470214639452816414062909567333}{1937241596269246950012279629010020801} a^{17} + \frac{510664421576742477350920034715584656}{1937241596269246950012279629010020801} a^{16} + \frac{895557681424687989840559831349519367}{1937241596269246950012279629010020801} a^{15} + \frac{868669480551626348793025825591788206}{1937241596269246950012279629010020801} a^{14} - \frac{29185297293702504020240189691713498}{1937241596269246950012279629010020801} a^{13} - \frac{38936792822570410190940668987142373}{1937241596269246950012279629010020801} a^{12} + \frac{957328696569885974844776475483184892}{1937241596269246950012279629010020801} a^{11} + \frac{215009407993123050611488079828247494}{1937241596269246950012279629010020801} a^{10} + \frac{833630065775557041403369563501833660}{1937241596269246950012279629010020801} a^{9} - \frac{842250026827496860329419162208675286}{1937241596269246950012279629010020801} a^{8} + \frac{856228913224845795027569975612387734}{1937241596269246950012279629010020801} a^{7} - \frac{155210692243309308191083403913721914}{1937241596269246950012279629010020801} a^{6} - \frac{585455973881485437716186682179653343}{1937241596269246950012279629010020801} a^{5} + \frac{564947554888636235751159795148573952}{1937241596269246950012279629010020801} a^{4} + \frac{588347071266955156990105270042036492}{1937241596269246950012279629010020801} a^{3} - \frac{714935284755050384047538316204998329}{1937241596269246950012279629010020801} a^{2} - \frac{638294027830168657593628331890671422}{1937241596269246950012279629010020801} a - \frac{815843174617852587385576513008434500}{1937241596269246950012279629010020801}$

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:  $11$
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:  \( 1084536.40133 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T751:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n751 are not computed
Character table for t20n751 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.219503494144.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$