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Properties

Label 20.0.38028917784...0625.2
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 7^{10}\cdot 13^{10}$
Root discriminant $21.33$
Ramified primes $5, 7, 13$
Class number $4$
Class group $[4]$
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![121, 110, 469, 189, 1202, -312, 1802, -1380, 2371, -2708, 2931, -2747, 2301, -1685, 1097, -658, 352, -154, 49, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 49*x^18 - 154*x^17 + 352*x^16 - 658*x^15 + 1097*x^14 - 1685*x^13 + 2301*x^12 - 2747*x^11 + 2931*x^10 - 2708*x^9 + 2371*x^8 - 1380*x^7 + 1802*x^6 - 312*x^5 + 1202*x^4 + 189*x^3 + 469*x^2 + 110*x + 121)
 
gp: K = bnfinit(x^20 - 10*x^19 + 49*x^18 - 154*x^17 + 352*x^16 - 658*x^15 + 1097*x^14 - 1685*x^13 + 2301*x^12 - 2747*x^11 + 2931*x^10 - 2708*x^9 + 2371*x^8 - 1380*x^7 + 1802*x^6 - 312*x^5 + 1202*x^4 + 189*x^3 + 469*x^2 + 110*x + 121, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 49 x^{18} - 154 x^{17} + 352 x^{16} - 658 x^{15} + 1097 x^{14} - 1685 x^{13} + 2301 x^{12} - 2747 x^{11} + 2931 x^{10} - 2708 x^{9} + 2371 x^{8} - 1380 x^{7} + 1802 x^{6} - 312 x^{5} + 1202 x^{4} + 189 x^{3} + 469 x^{2} + 110 x + 121 \)

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:  \(380289177849714310556640625=5^{10}\cdot 7^{10}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.33$
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:  $5, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{91} a^{16} + \frac{3}{13} a^{15} + \frac{11}{91} a^{14} - \frac{5}{13} a^{13} - \frac{12}{91} a^{12} + \frac{1}{13} a^{11} - \frac{44}{91} a^{10} + \frac{5}{13} a^{9} + \frac{3}{13} a^{7} - \frac{33}{91} a^{6} + \frac{5}{13} a^{5} - \frac{36}{91} a^{4} + \frac{4}{13} a^{3} + \frac{2}{91} a^{2} + \frac{6}{13} a + \frac{22}{91}$, $\frac{1}{1001} a^{17} + \frac{3}{1001} a^{16} + \frac{270}{1001} a^{15} + \frac{404}{1001} a^{14} - \frac{201}{1001} a^{13} + \frac{12}{91} a^{12} + \frac{12}{1001} a^{11} + \frac{9}{91} a^{10} + \frac{53}{143} a^{9} + \frac{42}{143} a^{8} + \frac{135}{1001} a^{7} - \frac{190}{1001} a^{6} + \frac{62}{1001} a^{5} - \frac{32}{77} a^{4} + \frac{4}{91} a^{3} - \frac{267}{1001} a^{2} + \frac{85}{1001} a - \frac{36}{91}$, $\frac{1}{1001} a^{18} - \frac{3}{1001} a^{16} + \frac{8}{143} a^{15} - \frac{313}{1001} a^{14} - \frac{5}{143} a^{13} - \frac{219}{1001} a^{12} + \frac{31}{143} a^{11} - \frac{46}{143} a^{10} - \frac{7}{143} a^{9} + \frac{254}{1001} a^{8} - \frac{19}{143} a^{7} + \frac{335}{1001} a^{6} + \frac{24}{143} a^{5} - \frac{215}{1001} a^{4} + \frac{31}{143} a^{3} + \frac{358}{1001} a^{2} + \frac{3}{11} a + \frac{5}{13}$, $\frac{1}{3004911568577743161337} a^{19} + \frac{885712432046709206}{3004911568577743161337} a^{18} + \frac{909409627261228985}{3004911568577743161337} a^{17} - \frac{3473845652699156759}{3004911568577743161337} a^{16} - \frac{396307476057398796053}{3004911568577743161337} a^{15} + \frac{415527381611265255194}{3004911568577743161337} a^{14} + \frac{461698864064176403941}{3004911568577743161337} a^{13} - \frac{570289200194473747062}{3004911568577743161337} a^{12} - \frac{967144425156608732465}{3004911568577743161337} a^{11} + \frac{354889547501867614909}{3004911568577743161337} a^{10} - \frac{1192066683964976681986}{3004911568577743161337} a^{9} - \frac{552219607603133072347}{3004911568577743161337} a^{8} - \frac{626165506089415171582}{3004911568577743161337} a^{7} + \frac{784714026689036008625}{3004911568577743161337} a^{6} + \frac{1372592419654100680941}{3004911568577743161337} a^{5} - \frac{1454969100527797790543}{3004911568577743161337} a^{4} - \frac{73524737333479832168}{231147043736749473949} a^{3} - \frac{1281403742239271177687}{3004911568577743161337} a^{2} + \frac{51093539197274942373}{273173778961613014667} a - \frac{128967445104510282663}{273173778961613014667}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( 37119.2526304 \)
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

\(\Q(\sqrt{-455}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{13}, \sqrt{-35})\), 5.1.207025.1 x5, 10.0.19501004534375.1, 10.0.1500077271875.1 x5, 10.2.557171558125.1 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 ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$