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Properties

Label 20.0.13350881126...0625.2
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 31^{10}$
Root discriminant $57.05$
Ramified primes $3, 5, 7, 31$
Class number $1936$ (GRH)
Class group $[22, 88]$ (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![18225, -91125, 224100, -221220, 19266, 48357, 244207, -587336, 653452, -481338, 279153, -127455, 53719, -19235, 6553, -1512, 462, -84, 37, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 37*x^18 - 84*x^17 + 462*x^16 - 1512*x^15 + 6553*x^14 - 19235*x^13 + 53719*x^12 - 127455*x^11 + 279153*x^10 - 481338*x^9 + 653452*x^8 - 587336*x^7 + 244207*x^6 + 48357*x^5 + 19266*x^4 - 221220*x^3 + 224100*x^2 - 91125*x + 18225)
 
gp: K = bnfinit(x^20 - 2*x^19 + 37*x^18 - 84*x^17 + 462*x^16 - 1512*x^15 + 6553*x^14 - 19235*x^13 + 53719*x^12 - 127455*x^11 + 279153*x^10 - 481338*x^9 + 653452*x^8 - 587336*x^7 + 244207*x^6 + 48357*x^5 + 19266*x^4 - 221220*x^3 + 224100*x^2 - 91125*x + 18225, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 37 x^{18} - 84 x^{17} + 462 x^{16} - 1512 x^{15} + 6553 x^{14} - 19235 x^{13} + 53719 x^{12} - 127455 x^{11} + 279153 x^{10} - 481338 x^{9} + 653452 x^{8} - 587336 x^{7} + 244207 x^{6} + 48357 x^{5} + 19266 x^{4} - 221220 x^{3} + 224100 x^{2} - 91125 x + 18225 \)

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:  \(133508811261782548210805849619140625=3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 31^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.05$
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:  $3, 5, 7, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{45} a^{9} - \frac{1}{15} a^{8} - \frac{2}{15} a^{7} + \frac{1}{15} a^{6} + \frac{1}{5} a^{5} + \frac{1}{15} a^{4} + \frac{11}{45} a^{3} - \frac{1}{15} a^{2} - \frac{1}{3} a$, $\frac{1}{45} a^{10} + \frac{1}{15} a^{6} + \frac{1}{9} a^{4} - \frac{1}{5} a^{2}$, $\frac{1}{45} a^{11} + \frac{1}{15} a^{7} + \frac{1}{9} a^{5} - \frac{1}{5} a^{3}$, $\frac{1}{45} a^{12} + \frac{1}{15} a^{8} + \frac{1}{9} a^{6} - \frac{1}{5} a^{4}$, $\frac{1}{135} a^{13} - \frac{1}{135} a^{12} - \frac{1}{135} a^{11} + \frac{1}{135} a^{10} + \frac{2}{45} a^{8} - \frac{2}{27} a^{7} + \frac{19}{135} a^{6} + \frac{64}{135} a^{5} + \frac{7}{27} a^{4} - \frac{1}{15} a^{3} - \frac{1}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{945} a^{14} + \frac{1}{135} a^{12} - \frac{1}{105} a^{11} + \frac{4}{945} a^{10} - \frac{7}{135} a^{8} - \frac{8}{105} a^{7} + \frac{74}{945} a^{6} - \frac{2}{21} a^{5} + \frac{437}{945} a^{4} - \frac{26}{105} a^{3} + \frac{157}{315} a^{2} + \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{945} a^{15} - \frac{2}{945} a^{12} - \frac{2}{189} a^{11} - \frac{1}{135} a^{10} - \frac{1}{135} a^{9} + \frac{5}{63} a^{8} + \frac{16}{105} a^{7} - \frac{97}{945} a^{6} - \frac{368}{945} a^{5} - \frac{353}{945} a^{4} - \frac{26}{63} a^{3} + \frac{23}{315} a^{2} - \frac{3}{7} a$, $\frac{1}{14175} a^{16} - \frac{1}{14175} a^{15} - \frac{2}{14175} a^{14} + \frac{26}{14175} a^{13} + \frac{34}{14175} a^{12} - \frac{4}{2025} a^{11} - \frac{106}{14175} a^{10} + \frac{103}{14175} a^{9} - \frac{106}{14175} a^{8} + \frac{1954}{14175} a^{7} - \frac{412}{14175} a^{6} - \frac{1079}{14175} a^{5} - \frac{943}{4725} a^{4} + \frac{86}{189} a^{3} - \frac{59}{315} a^{2} - \frac{2}{21} a - \frac{1}{7}$, $\frac{1}{14175} a^{17} - \frac{1}{4725} a^{15} - \frac{2}{4725} a^{14} - \frac{1}{315} a^{13} - \frac{11}{1575} a^{12} - \frac{74}{14175} a^{11} + \frac{29}{4725} a^{10} - \frac{1}{4725} a^{9} - \frac{97}{675} a^{8} - \frac{34}{525} a^{7} - \frac{4}{1575} a^{6} - \frac{53}{14175} a^{5} - \frac{2288}{4725} a^{4} - \frac{41}{135} a^{3} + \frac{94}{315} a^{2} - \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{14175} a^{18} + \frac{2}{4725} a^{15} - \frac{2}{4725} a^{14} - \frac{1}{675} a^{13} - \frac{2}{14175} a^{12} + \frac{26}{4725} a^{11} + \frac{23}{4725} a^{10} + \frac{19}{4725} a^{9} + \frac{173}{4725} a^{8} + \frac{1}{675} a^{7} + \frac{901}{14175} a^{6} + \frac{1738}{4725} a^{5} + \frac{2311}{4725} a^{4} - \frac{29}{315} a^{3} - \frac{1}{45} a^{2} + \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{102555765111877752965498864042625} a^{19} - \frac{2532752131798114886085555739}{102555765111877752965498864042625} a^{18} + \frac{42198996751513412276878831}{6837051007458516864366590936175} a^{17} - \frac{51547062160431991175021639}{4883607862470369188833279240125} a^{16} - \frac{707364477362640174000674629}{2279017002486172288122196978725} a^{15} - \frac{17497104073026138980934665419}{34185255037292584321832954680875} a^{14} + \frac{304052601818175571408406058622}{102555765111877752965498864042625} a^{13} - \frac{34218263691190056019937502772}{14650823587411107566499837720375} a^{12} + \frac{45053702369926391101704513979}{34185255037292584321832954680875} a^{11} - \frac{211025656303182468001552376363}{34185255037292584321832954680875} a^{10} + \frac{28004702069324896542465564394}{11395085012430861440610984893625} a^{9} - \frac{302357420960518295461456991986}{6837051007458516864366590936175} a^{8} - \frac{1723825893077296436132335055423}{102555765111877752965498864042625} a^{7} + \frac{2456103448151215536645850871677}{20511153022375550593099772808525} a^{6} - \frac{477115978644986128704415969637}{11395085012430861440610984893625} a^{5} + \frac{1471979924165698646167014778921}{34185255037292584321832954680875} a^{4} + \frac{512616794095308347859409831942}{2279017002486172288122196978725} a^{3} - \frac{14730217958869872490950934048}{151934466832411485874813131915} a^{2} - \frac{410648914411070546669635781}{50644822277470495291604377305} a - \frac{134678752527042648772750523}{2411658203689071204362113205}$

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

Class group and class number

$C_{22}\times C_{88}$, which has order $1936$ (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:  \( 255586981.075 \) (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

\(\Q(\sqrt{-3255}) \), \(\Q(\sqrt{-155}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{21}, \sqrt{-155})\), 5.1.10595025.1 x5, 10.0.365388575713284375.1, 10.0.17399455986346875.1 x5, 10.2.2357345649763125.1 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{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$$[\ ]_{2}^{2}$
$31$31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$