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Properties

Label 20.0.20000000000...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{24}\cdot 5^{23}$
Root discriminant $14.62$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -125, 300, -500, 700, -875, 1000, -1075, 1050, -850, 515, -200, 40, -25, 40, -25, 10, -10, 10, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 10*x^18 - 10*x^17 + 10*x^16 - 25*x^15 + 40*x^14 - 25*x^13 + 40*x^12 - 200*x^11 + 515*x^10 - 850*x^9 + 1050*x^8 - 1075*x^7 + 1000*x^6 - 875*x^5 + 700*x^4 - 500*x^3 + 300*x^2 - 125*x + 25)
 
gp: K = bnfinit(x^20 - 5*x^19 + 10*x^18 - 10*x^17 + 10*x^16 - 25*x^15 + 40*x^14 - 25*x^13 + 40*x^12 - 200*x^11 + 515*x^10 - 850*x^9 + 1050*x^8 - 1075*x^7 + 1000*x^6 - 875*x^5 + 700*x^4 - 500*x^3 + 300*x^2 - 125*x + 25, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 10 x^{18} - 10 x^{17} + 10 x^{16} - 25 x^{15} + 40 x^{14} - 25 x^{13} + 40 x^{12} - 200 x^{11} + 515 x^{10} - 850 x^{9} + 1050 x^{8} - 1075 x^{7} + 1000 x^{6} - 875 x^{5} + 700 x^{4} - 500 x^{3} + 300 x^{2} - 125 x + 25 \)

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:  \(200000000000000000000000=2^{24}\cdot 5^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.62$
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:  $2, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10}$, $\frac{1}{5} a^{11}$, $\frac{1}{5} a^{12}$, $\frac{1}{5} a^{13}$, $\frac{1}{5} a^{14}$, $\frac{1}{5} a^{15}$, $\frac{1}{75} a^{16} - \frac{1}{15} a^{15} - \frac{1}{15} a^{14} - \frac{1}{15} a^{13} - \frac{1}{15} a^{12} + \frac{1}{15} a^{11} - \frac{1}{15} a^{10} - \frac{1}{3} a^{9} - \frac{2}{5} a^{8} - \frac{1}{3} a^{7} - \frac{1}{15} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{75} a^{17} - \frac{1}{15} a^{12} + \frac{1}{15} a^{11} - \frac{1}{15} a^{10} - \frac{1}{15} a^{9} - \frac{1}{3} a^{8} + \frac{4}{15} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{26925} a^{18} - \frac{29}{26925} a^{17} - \frac{104}{26925} a^{16} - \frac{118}{5385} a^{15} + \frac{296}{5385} a^{14} + \frac{157}{5385} a^{13} - \frac{118}{5385} a^{12} - \frac{488}{5385} a^{11} + \frac{106}{1795} a^{10} + \frac{1324}{5385} a^{9} + \frac{1178}{5385} a^{8} + \frac{2684}{5385} a^{7} - \frac{2416}{5385} a^{6} + \frac{98}{359} a^{5} - \frac{113}{1077} a^{4} + \frac{400}{1077} a^{3} - \frac{253}{1077} a^{2} + \frac{42}{359} a + \frac{173}{359}$, $\frac{1}{2396325} a^{19} + \frac{29}{2396325} a^{18} - \frac{3228}{798775} a^{17} + \frac{827}{159755} a^{16} - \frac{14987}{159755} a^{15} - \frac{44782}{479265} a^{14} + \frac{44888}{479265} a^{13} - \frac{22769}{479265} a^{12} + \frac{2279}{159755} a^{11} - \frac{19363}{479265} a^{10} + \frac{54351}{159755} a^{9} + \frac{1774}{5385} a^{8} - \frac{61162}{159755} a^{7} - \frac{21844}{95853} a^{6} - \frac{23269}{95853} a^{5} - \frac{33079}{95853} a^{4} + \frac{18280}{95853} a^{3} + \frac{7955}{31951} a^{2} + \frac{30803}{95853} a - \frac{3608}{31951}$

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:  \( \frac{200932}{159755} a^{19} - \frac{13016039}{2396325} a^{18} + \frac{1408004}{159755} a^{17} - \frac{1013306}{159755} a^{16} + \frac{254664}{31951} a^{15} - \frac{4134648}{159755} a^{14} + \frac{15493316}{479265} a^{13} - \frac{797455}{95853} a^{12} + \frac{4198858}{95853} a^{11} - \frac{106333924}{479265} a^{10} + \frac{47356628}{95853} a^{9} - \frac{3892834}{5385} a^{8} + \frac{25885224}{31951} a^{7} - \frac{24789995}{31951} a^{6} + \frac{67398476}{95853} a^{5} - \frac{57307868}{95853} a^{4} + \frac{14454758}{31951} a^{3} - \frac{29180228}{95853} a^{2} + \frac{15200710}{95853} a - \frac{1326305}{31951} \) (order $10$)
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:  \( 13801.9175678 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.200000.1, 10.2.200000000000.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
5Data not computed