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Properties

Label 20.0.40319878008...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{15}\cdot 17^{10}$
Root discriminant $24.00$
Ramified primes $2, 5, 17$
Class number $2$
Class group $[2]$
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![25, 175, 575, 1025, 990, 325, -335, -335, 236, 203, 169, -677, 262, -153, 399, -221, -36, 51, -3, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 3*x^18 + 51*x^17 - 36*x^16 - 221*x^15 + 399*x^14 - 153*x^13 + 262*x^12 - 677*x^11 + 169*x^10 + 203*x^9 + 236*x^8 - 335*x^7 - 335*x^6 + 325*x^5 + 990*x^4 + 1025*x^3 + 575*x^2 + 175*x + 25)
 
gp: K = bnfinit(x^20 - 5*x^19 - 3*x^18 + 51*x^17 - 36*x^16 - 221*x^15 + 399*x^14 - 153*x^13 + 262*x^12 - 677*x^11 + 169*x^10 + 203*x^9 + 236*x^8 - 335*x^7 - 335*x^6 + 325*x^5 + 990*x^4 + 1025*x^3 + 575*x^2 + 175*x + 25, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 3 x^{18} + 51 x^{17} - 36 x^{16} - 221 x^{15} + 399 x^{14} - 153 x^{13} + 262 x^{12} - 677 x^{11} + 169 x^{10} + 203 x^{9} + 236 x^{8} - 335 x^{7} - 335 x^{6} + 325 x^{5} + 990 x^{4} + 1025 x^{3} + 575 x^{2} + 175 x + 25 \)

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:  \(4031987800898000000000000000=2^{16}\cdot 5^{15}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.00$
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:  $2, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{8} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{9} + \frac{2}{5} a^{4}$, $\frac{1}{5} a^{15} - \frac{2}{5} a^{10} + \frac{2}{5} a^{5}$, $\frac{1}{5} a^{16} - \frac{2}{5} a^{11} + \frac{2}{5} a^{6}$, $\frac{1}{5} a^{17} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4}$, $\frac{1}{475} a^{18} - \frac{47}{475} a^{17} - \frac{34}{475} a^{16} - \frac{46}{475} a^{15} - \frac{44}{475} a^{14} - \frac{3}{475} a^{13} + \frac{3}{95} a^{12} - \frac{88}{475} a^{11} + \frac{103}{475} a^{10} - \frac{12}{25} a^{9} - \frac{24}{95} a^{8} - \frac{77}{475} a^{7} + \frac{22}{95} a^{6} - \frac{4}{19} a^{5} - \frac{36}{95} a^{4} - \frac{28}{95} a^{3} + \frac{1}{19} a^{2} - \frac{2}{19} a + \frac{2}{19}$, $\frac{1}{114300026230909533025} a^{19} + \frac{7058570822741704}{22860005246181906605} a^{18} - \frac{7993715330559387348}{114300026230909533025} a^{17} + \frac{8654399593093092631}{114300026230909533025} a^{16} + \frac{10865690259891822609}{114300026230909533025} a^{15} + \frac{8304266903010929469}{114300026230909533025} a^{14} + \frac{7278006487408128729}{114300026230909533025} a^{13} + \frac{9218280518322146882}{114300026230909533025} a^{12} + \frac{1975981405474657467}{10390911475537230275} a^{11} - \frac{23052934488878861837}{114300026230909533025} a^{10} + \frac{10133708930698039029}{114300026230909533025} a^{9} - \frac{46110812951327474062}{114300026230909533025} a^{8} + \frac{1963752051356600464}{6015790854258396475} a^{7} - \frac{558020539300844703}{22860005246181906605} a^{6} - \frac{7842812998391431448}{22860005246181906605} a^{5} - \frac{7318206145924743896}{22860005246181906605} a^{4} + \frac{7856617336212350379}{22860005246181906605} a^{3} + \frac{704375244550844807}{4572001049236381321} a^{2} - \frac{677639372599180857}{4572001049236381321} a - \frac{2129276002756734725}{4572001049236381321}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( 246762.21333 \)
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

\(\Q(\sqrt{5}) \), 4.0.36125.1, 5.1.578000.2 x5, 10.2.1670420000000.2 x5

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

Sibling fields

Degree 5 sibling: 5.1.578000.2
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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
2Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$17$17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$