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Properties

Label 20.0.44136757656...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{22}\cdot 7^{10}$
Root discriminant $27.05$
Ramified primes $2, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59049, -196830, 360855, -459270, 433755, -303750, 140940, -13770, -48915, 57780, -40229, 19260, -5435, -510, 1740, -1250, 595, -210, 55, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 55*x^18 - 210*x^17 + 595*x^16 - 1250*x^15 + 1740*x^14 - 510*x^13 - 5435*x^12 + 19260*x^11 - 40229*x^10 + 57780*x^9 - 48915*x^8 - 13770*x^7 + 140940*x^6 - 303750*x^5 + 433755*x^4 - 459270*x^3 + 360855*x^2 - 196830*x + 59049)
 
gp: K = bnfinit(x^20 - 10*x^19 + 55*x^18 - 210*x^17 + 595*x^16 - 1250*x^15 + 1740*x^14 - 510*x^13 - 5435*x^12 + 19260*x^11 - 40229*x^10 + 57780*x^9 - 48915*x^8 - 13770*x^7 + 140940*x^6 - 303750*x^5 + 433755*x^4 - 459270*x^3 + 360855*x^2 - 196830*x + 59049, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 55 x^{18} - 210 x^{17} + 595 x^{16} - 1250 x^{15} + 1740 x^{14} - 510 x^{13} - 5435 x^{12} + 19260 x^{11} - 40229 x^{10} + 57780 x^{9} - 48915 x^{8} - 13770 x^{7} + 140940 x^{6} - 303750 x^{5} + 433755 x^{4} - 459270 x^{3} + 360855 x^{2} - 196830 x + 59049 \)

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:  \(44136757656250000000000000000=2^{16}\cdot 5^{22}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.05$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{6} a^{9} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{18} a^{12} - \frac{1}{18} a^{11} + \frac{1}{18} a^{10} + \frac{1}{3} a^{9} + \frac{1}{18} a^{8} + \frac{1}{18} a^{7} + \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{18} a^{4} - \frac{1}{2} a^{3} + \frac{1}{18} a^{2}$, $\frac{1}{54} a^{13} - \frac{1}{54} a^{12} + \frac{1}{54} a^{11} + \frac{1}{9} a^{10} - \frac{17}{54} a^{9} + \frac{1}{54} a^{8} + \frac{1}{18} a^{7} + \frac{2}{9} a^{6} + \frac{19}{54} a^{5} - \frac{1}{6} a^{4} - \frac{17}{54} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{162} a^{14} - \frac{1}{162} a^{13} + \frac{1}{162} a^{12} + \frac{1}{27} a^{11} + \frac{37}{162} a^{10} - \frac{53}{162} a^{9} + \frac{1}{54} a^{8} - \frac{7}{27} a^{7} - \frac{35}{162} a^{6} - \frac{7}{18} a^{5} - \frac{17}{162} a^{4} + \frac{2}{9} a^{3} - \frac{2}{9} a^{2}$, $\frac{1}{486} a^{15} - \frac{1}{486} a^{14} + \frac{1}{486} a^{13} + \frac{1}{81} a^{12} + \frac{37}{486} a^{11} + \frac{109}{486} a^{10} + \frac{55}{162} a^{9} + \frac{20}{81} a^{8} + \frac{127}{486} a^{7} - \frac{25}{54} a^{6} + \frac{145}{486} a^{5} - \frac{7}{27} a^{4} + \frac{7}{27} a^{3}$, $\frac{1}{1458} a^{16} - \frac{1}{1458} a^{15} + \frac{1}{1458} a^{14} + \frac{1}{243} a^{13} + \frac{37}{1458} a^{12} + \frac{109}{1458} a^{11} + \frac{55}{486} a^{10} - \frac{61}{243} a^{9} - \frac{359}{1458} a^{8} - \frac{25}{162} a^{7} + \frac{145}{1458} a^{6} + \frac{20}{81} a^{5} + \frac{34}{81} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{4374} a^{17} - \frac{1}{4374} a^{16} + \frac{1}{4374} a^{15} + \frac{1}{729} a^{14} + \frac{37}{4374} a^{13} + \frac{109}{4374} a^{12} + \frac{55}{1458} a^{11} + \frac{182}{729} a^{10} + \frac{1099}{4374} a^{9} - \frac{187}{486} a^{8} - \frac{1313}{4374} a^{7} - \frac{61}{243} a^{6} + \frac{115}{243} a^{5} - \frac{2}{9} a^{4} - \frac{1}{3} a^{3} - \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{13633758} a^{18} + \frac{247}{6816879} a^{17} - \frac{166}{6816879} a^{16} - \frac{1273}{4544586} a^{15} + \frac{15589}{13633758} a^{14} + \frac{29489}{6816879} a^{13} - \frac{55618}{2272293} a^{12} + \frac{118712}{2272293} a^{11} - \frac{1535899}{6816879} a^{10} + \frac{75793}{504954} a^{9} + \frac{210146}{6816879} a^{8} + \frac{235612}{757431} a^{7} + \frac{419189}{1514862} a^{6} - \frac{210613}{504954} a^{5} + \frac{347}{9351} a^{4} - \frac{5897}{18702} a^{3} + \frac{2785}{6234} a^{2} + \frac{443}{6234} a - \frac{518}{1039}$, $\frac{1}{40901274} a^{19} - \frac{1}{40901274} a^{18} - \frac{868}{20450637} a^{17} + \frac{259}{6816879} a^{16} - \frac{19507}{20450637} a^{15} + \frac{28945}{40901274} a^{14} + \frac{56546}{6816879} a^{13} - \frac{304919}{13633758} a^{12} - \frac{1701737}{40901274} a^{11} + \frac{109642}{2272293} a^{10} - \frac{2065391}{40901274} a^{9} + \frac{609458}{2272293} a^{8} + \frac{721112}{2272293} a^{7} - \frac{450577}{1514862} a^{6} + \frac{10013}{56106} a^{5} - \frac{18667}{84159} a^{4} + \frac{4325}{18702} a^{3} - \frac{7691}{18702} a^{2} - \frac{467}{2078} a - \frac{495}{2078}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 1633832.53547 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

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

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-7})\), 5.5.2450000.1, 10.0.210087500000000.1, 10.0.42017500000000.1, 10.10.30012500000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
5Data not computed
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$