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Properties

Label 20.0.55612146387...5625.2
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 7^{10}\cdot 17^{10}$
Root discriminant $24.39$
Ramified primes $5, 7, 17$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10031, -31710, 42435, -12998, -32318, 51815, -34663, 6378, 10464, -12770, 8769, -4636, 2616, -1626, 825, -279, 99, -48, 22, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 22*x^18 - 48*x^17 + 99*x^16 - 279*x^15 + 825*x^14 - 1626*x^13 + 2616*x^12 - 4636*x^11 + 8769*x^10 - 12770*x^9 + 10464*x^8 + 6378*x^7 - 34663*x^6 + 51815*x^5 - 32318*x^4 - 12998*x^3 + 42435*x^2 - 31710*x + 10031)
 
gp: K = bnfinit(x^20 - 6*x^19 + 22*x^18 - 48*x^17 + 99*x^16 - 279*x^15 + 825*x^14 - 1626*x^13 + 2616*x^12 - 4636*x^11 + 8769*x^10 - 12770*x^9 + 10464*x^8 + 6378*x^7 - 34663*x^6 + 51815*x^5 - 32318*x^4 - 12998*x^3 + 42435*x^2 - 31710*x + 10031, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 22 x^{18} - 48 x^{17} + 99 x^{16} - 279 x^{15} + 825 x^{14} - 1626 x^{13} + 2616 x^{12} - 4636 x^{11} + 8769 x^{10} - 12770 x^{9} + 10464 x^{8} + 6378 x^{7} - 34663 x^{6} + 51815 x^{5} - 32318 x^{4} - 12998 x^{3} + 42435 x^{2} - 31710 x + 10031 \)

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:  \(5561214638787231316416015625=5^{10}\cdot 7^{10}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.39$
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:  $5, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{11} + \frac{3}{7} a^{10} - \frac{2}{7} a^{9} - \frac{3}{7} a^{8} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2}$, $\frac{1}{7} a^{13} + \frac{2}{7} a^{11} + \frac{2}{7} a^{10} - \frac{1}{7} a^{9} + \frac{3}{7} a^{8} - \frac{1}{7} a^{7} - \frac{2}{7} a^{5} - \frac{2}{7} a^{4} + \frac{1}{7} a^{3} - \frac{3}{7} a^{2}$, $\frac{1}{7} a^{14} - \frac{2}{7} a^{8} + \frac{1}{7} a^{2}$, $\frac{1}{49} a^{15} - \frac{3}{49} a^{14} - \frac{3}{49} a^{13} + \frac{3}{49} a^{12} - \frac{3}{49} a^{11} - \frac{18}{49} a^{10} + \frac{2}{49} a^{9} + \frac{16}{49} a^{8} + \frac{10}{49} a^{7} + \frac{18}{49} a^{6} - \frac{11}{49} a^{5} + \frac{4}{49} a^{4} + \frac{11}{49} a^{3} - \frac{13}{49} a^{2} - \frac{2}{7}$, $\frac{1}{833} a^{16} + \frac{2}{833} a^{15} - \frac{53}{833} a^{14} + \frac{37}{833} a^{13} + \frac{47}{833} a^{12} + \frac{296}{833} a^{11} - \frac{32}{833} a^{10} + \frac{201}{833} a^{9} - \frac{337}{833} a^{8} + \frac{166}{833} a^{7} + \frac{17}{49} a^{6} + \frac{61}{833} a^{5} - \frac{319}{833} a^{4} - \frac{33}{119} a^{3} + \frac{397}{833} a^{2} - \frac{2}{119} a - \frac{10}{119}$, $\frac{1}{10829} a^{17} - \frac{108}{10829} a^{15} - \frac{180}{10829} a^{14} + \frac{1}{1547} a^{13} - \frac{6}{119} a^{12} + \frac{362}{10829} a^{11} + \frac{594}{1547} a^{10} + \frac{3800}{10829} a^{9} + \frac{1571}{10829} a^{8} - \frac{657}{1547} a^{7} + \frac{594}{1547} a^{6} + \frac{4285}{10829} a^{5} + \frac{556}{1547} a^{4} + \frac{2321}{10829} a^{3} - \frac{383}{10829} a^{2} - \frac{125}{1547} a - \frac{711}{1547}$, $\frac{1}{24181157} a^{18} - \frac{151}{24181157} a^{17} + \frac{230}{24181157} a^{16} + \frac{30285}{24181157} a^{15} - \frac{1076942}{24181157} a^{14} - \frac{147428}{3454451} a^{13} - \frac{93}{24181157} a^{12} - \frac{2126}{2198287} a^{11} - \frac{9191110}{24181157} a^{10} - \frac{6126973}{24181157} a^{9} + \frac{63073}{2198287} a^{8} + \frac{124}{539} a^{7} + \frac{1469518}{24181157} a^{6} + \frac{10372225}{24181157} a^{5} + \frac{11247879}{24181157} a^{4} + \frac{3441441}{24181157} a^{3} + \frac{2631868}{24181157} a^{2} + \frac{1615539}{3454451} a - \frac{951736}{3454451}$, $\frac{1}{4091463595959552897758823149} a^{19} - \frac{1237958783873587163}{371951235996322990705347559} a^{18} + \frac{3803781067189019560311}{141084951584812168888235281} a^{17} + \frac{48039186377786601169113}{141084951584812168888235281} a^{16} + \frac{37127031906002891965762690}{4091463595959552897758823149} a^{15} + \frac{6523148846086514010445904}{240674329174091346926989597} a^{14} + \frac{2791666135831849917396255}{371951235996322990705347559} a^{13} + \frac{61281269031316368985337523}{4091463595959552897758823149} a^{12} + \frac{133025346356371396945718693}{314727968919965607519909473} a^{11} - \frac{459082651203553989221926450}{4091463595959552897758823149} a^{10} - \frac{1731509295461586359811669867}{4091463595959552897758823149} a^{9} - \frac{120190528148764190195268661}{314727968919965607519909473} a^{8} + \frac{1339357211562205467848465166}{4091463595959552897758823149} a^{7} + \frac{122569592825671703414791709}{314727968919965607519909473} a^{6} + \frac{1914669186699195146409139674}{4091463595959552897758823149} a^{5} - \frac{1436768425091447930411051321}{4091463595959552897758823149} a^{4} - \frac{16525575118985131257236333}{177889721563458821641687963} a^{3} + \frac{44582081840398885226230443}{141084951584812168888235281} a^{2} - \frac{24956831670874787416606868}{53135890856617570100763937} a + \frac{14393247590165161686693762}{584494799422793271108403307}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 373616.54629 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-119}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{-35}, \sqrt{85})\), 5.1.14161.1 x5, 10.0.23863536599.2, 10.0.4386679521875.2 x5, 10.2.10653364553125.1 x5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$