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Properties

Label 20.0.21852734616...7248.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 37^{15}$
Root discriminant $26.12$
Ramified primes $2, 37$
Class number $1$
Class group Trivial
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1381, 1177, -876, -1854, 4070, -3773, 1310, -830, 3457, -7762, 11680, -13058, 11428, -8044, 4622, -2176, 833, -255, 60, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 60*x^18 - 255*x^17 + 833*x^16 - 2176*x^15 + 4622*x^14 - 8044*x^13 + 11428*x^12 - 13058*x^11 + 11680*x^10 - 7762*x^9 + 3457*x^8 - 830*x^7 + 1310*x^6 - 3773*x^5 + 4070*x^4 - 1854*x^3 - 876*x^2 + 1177*x + 1381)
 
gp: K = bnfinit(x^20 - 10*x^19 + 60*x^18 - 255*x^17 + 833*x^16 - 2176*x^15 + 4622*x^14 - 8044*x^13 + 11428*x^12 - 13058*x^11 + 11680*x^10 - 7762*x^9 + 3457*x^8 - 830*x^7 + 1310*x^6 - 3773*x^5 + 4070*x^4 - 1854*x^3 - 876*x^2 + 1177*x + 1381, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 60 x^{18} - 255 x^{17} + 833 x^{16} - 2176 x^{15} + 4622 x^{14} - 8044 x^{13} + 11428 x^{12} - 13058 x^{11} + 11680 x^{10} - 7762 x^{9} + 3457 x^{8} - 830 x^{7} + 1310 x^{6} - 3773 x^{5} + 4070 x^{4} - 1854 x^{3} - 876 x^{2} + 1177 x + 1381 \)

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:  \(21852734616490167965351477248=2^{16}\cdot 37^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.12$
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:  $2, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{4}{9} a^{5} - \frac{1}{3} a^{4} - \frac{1}{9} a^{3} - \frac{4}{9} a^{2} + \frac{1}{9} a - \frac{1}{9}$, $\frac{1}{9} a^{11} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{1}{9} a^{5} - \frac{4}{9} a^{4} - \frac{1}{3} a^{3} - \frac{1}{9} a^{2} - \frac{2}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{4}{9} a^{5} - \frac{1}{3} a^{4} - \frac{1}{9} a^{3} - \frac{2}{9} a^{2} + \frac{1}{9} a$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{2}{9} a^{5} - \frac{4}{9} a^{4} - \frac{4}{9} a^{3} - \frac{1}{9} a^{2} - \frac{4}{9} a + \frac{1}{9}$, $\frac{1}{27} a^{14} - \frac{1}{27} a^{13} - \frac{1}{27} a^{12} + \frac{1}{27} a^{11} - \frac{1}{27} a^{10} + \frac{2}{27} a^{9} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{7}{27} a^{5} - \frac{13}{27} a^{4} + \frac{2}{27} a^{3} - \frac{13}{27} a^{2} - \frac{2}{27} a - \frac{2}{27}$, $\frac{1}{2241} a^{15} + \frac{34}{2241} a^{14} - \frac{26}{747} a^{13} - \frac{64}{2241} a^{12} + \frac{103}{2241} a^{11} - \frac{1}{249} a^{10} - \frac{311}{2241} a^{9} + \frac{46}{747} a^{8} - \frac{70}{747} a^{7} - \frac{41}{2241} a^{6} + \frac{1111}{2241} a^{5} + \frac{2}{747} a^{4} - \frac{235}{747} a^{3} + \frac{56}{2241} a^{2} + \frac{11}{747} a - \frac{364}{2241}$, $\frac{1}{2241} a^{16} + \frac{11}{2241} a^{14} + \frac{98}{2241} a^{13} + \frac{38}{2241} a^{12} - \frac{25}{2241} a^{11} - \frac{5}{2241} a^{10} + \frac{5}{2241} a^{9} + \frac{26}{747} a^{8} - \frac{122}{2241} a^{7} - \frac{26}{249} a^{6} + \frac{827}{2241} a^{5} - \frac{6}{83} a^{4} - \frac{874}{2241} a^{3} - \frac{377}{2241} a^{2} + \frac{506}{2241} a - \frac{74}{2241}$, $\frac{1}{2241} a^{17} - \frac{1}{83} a^{14} - \frac{100}{2241} a^{13} - \frac{68}{2241} a^{12} + \frac{107}{2241} a^{11} + \frac{104}{2241} a^{10} + \frac{13}{2241} a^{9} - \frac{146}{2241} a^{8} + \frac{37}{249} a^{7} - \frac{8}{83} a^{6} - \frac{431}{2241} a^{5} - \frac{691}{2241} a^{4} - \frac{839}{2241} a^{3} + \frac{139}{2241} a^{2} - \frac{935}{2241} a - \frac{229}{2241}$, $\frac{1}{32178519} a^{18} - \frac{1}{3575391} a^{17} + \frac{5093}{32178519} a^{16} + \frac{2537}{32178519} a^{15} - \frac{522899}{32178519} a^{14} + \frac{59464}{3575391} a^{13} - \frac{279433}{32178519} a^{12} + \frac{1449715}{32178519} a^{11} + \frac{1235293}{32178519} a^{10} - \frac{3731899}{32178519} a^{9} - \frac{1517072}{10726173} a^{8} + \frac{1040900}{32178519} a^{7} - \frac{1254736}{10726173} a^{6} + \frac{10884442}{32178519} a^{5} + \frac{13442399}{32178519} a^{4} - \frac{100087}{3575391} a^{3} - \frac{7760971}{32178519} a^{2} + \frac{6398963}{32178519} a - \frac{15297601}{32178519}$, $\frac{1}{2670817077} a^{19} + \frac{32}{2670817077} a^{18} + \frac{263186}{2670817077} a^{17} + \frac{354940}{2670817077} a^{16} + \frac{26848}{296757453} a^{15} + \frac{872820}{98919151} a^{14} - \frac{2753791}{890272359} a^{13} - \frac{6000877}{2670817077} a^{12} - \frac{109423106}{2670817077} a^{11} - \frac{81842039}{2670817077} a^{10} - \frac{2440286}{890272359} a^{9} + \frac{329555810}{2670817077} a^{8} - \frac{285098143}{2670817077} a^{7} - \frac{256381621}{2670817077} a^{6} - \frac{891908149}{2670817077} a^{5} + \frac{53583079}{296757453} a^{4} - \frac{417318239}{890272359} a^{3} + \frac{258782735}{2670817077} a^{2} + \frac{136572878}{890272359} a + \frac{305602076}{2670817077}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( 1702148.10944 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{37}) \), 4.0.50653.1, 5.1.810448.1 x5, 10.2.24302560546048.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.810448.1
Degree 10 sibling: 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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
$37$37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$