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Properties

Label 20.4.33626538312...2249.1
Degree $20$
Signature $[4, 8]$
Discriminant $3^{10}\cdot 7^{10}\cdot 17^{10}$
Root discriminant $18.89$
Ramified primes $3, 7, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2\times D_5$ (as 20T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -67, 322, -612, 305, 947, -2634, 3655, -2975, 1036, 374, -409, -148, 450, -428, 283, -138, 47, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 47*x^18 - 138*x^17 + 283*x^16 - 428*x^15 + 450*x^14 - 148*x^13 - 409*x^12 + 374*x^11 + 1036*x^10 - 2975*x^9 + 3655*x^8 - 2634*x^7 + 947*x^6 + 305*x^5 - 612*x^4 + 322*x^3 - 67*x^2 + x + 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 47*x^18 - 138*x^17 + 283*x^16 - 428*x^15 + 450*x^14 - 148*x^13 - 409*x^12 + 374*x^11 + 1036*x^10 - 2975*x^9 + 3655*x^8 - 2634*x^7 + 947*x^6 + 305*x^5 - 612*x^4 + 322*x^3 - 67*x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 47 x^{18} - 138 x^{17} + 283 x^{16} - 428 x^{15} + 450 x^{14} - 148 x^{13} - 409 x^{12} + 374 x^{11} + 1036 x^{10} - 2975 x^{9} + 3655 x^{8} - 2634 x^{7} + 947 x^{6} + 305 x^{5} - 612 x^{4} + 322 x^{3} - 67 x^{2} + x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(33626538312268515533112249=3^{10}\cdot 7^{10}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.89$
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:  $3, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{11} - \frac{1}{7} a^{10} - \frac{3}{7} a^{9} - \frac{1}{7} a^{8} - \frac{2}{7} a^{7} + \frac{2}{7} a^{6} - \frac{3}{7} a^{5} - \frac{1}{7} a^{3} - \frac{1}{7} a^{2} + \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{13} - \frac{2}{7} a^{11} - \frac{2}{7} a^{10} + \frac{2}{7} a^{9} - \frac{1}{7} a^{8} - \frac{3}{7} a^{7} + \frac{2}{7} a^{6} + \frac{3}{7} a^{5} - \frac{1}{7} a^{4} + \frac{2}{7} a^{2} + \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{14} + \frac{2}{7} a^{8} - \frac{2}{7} a^{7} + \frac{1}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{15} + \frac{2}{7} a^{9} - \frac{2}{7} a^{8} + \frac{1}{7} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{7} a^{16} + \frac{2}{7} a^{10} - \frac{2}{7} a^{9} + \frac{1}{7} a^{4} - \frac{2}{7} a^{3} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{17} + \frac{2}{7} a^{11} - \frac{2}{7} a^{10} + \frac{1}{7} a^{5} - \frac{2}{7} a^{4} + \frac{1}{7} a^{3}$, $\frac{1}{50699467} a^{18} - \frac{9}{50699467} a^{17} - \frac{414041}{7242781} a^{16} + \frac{1458157}{50699467} a^{15} + \frac{1723915}{50699467} a^{14} + \frac{1301}{29323} a^{13} - \frac{2248522}{50699467} a^{12} - \frac{18209069}{50699467} a^{11} + \frac{5342430}{50699467} a^{10} + \frac{6766273}{50699467} a^{9} - \frac{10902657}{50699467} a^{8} - \frac{3823174}{50699467} a^{7} - \frac{24509865}{50699467} a^{6} - \frac{19565878}{50699467} a^{5} - \frac{23275856}{50699467} a^{4} - \frac{1865545}{50699467} a^{3} + \frac{3134961}{7242781} a^{2} + \frac{2628901}{50699467} a + \frac{11121605}{50699467}$, $\frac{1}{12826965151} a^{19} + \frac{9}{986689627} a^{18} + \frac{17175408}{1832423593} a^{17} - \frac{42942520}{1166087741} a^{16} + \frac{373764003}{12826965151} a^{15} - \frac{65908385}{1832423593} a^{14} + \frac{838873669}{12826965151} a^{13} + \frac{2673961}{12826965151} a^{12} - \frac{28996712}{1166087741} a^{11} + \frac{452920397}{1166087741} a^{10} + \frac{86663040}{557694137} a^{9} + \frac{4894690390}{12826965151} a^{8} + \frac{206199527}{986689627} a^{7} + \frac{238355954}{12826965151} a^{6} - \frac{2517547608}{12826965151} a^{5} - \frac{1362939924}{12826965151} a^{4} - \frac{353265945}{1832423593} a^{3} + \frac{1449478361}{12826965151} a^{2} - \frac{84960948}{12826965151} a - \frac{635834974}{1832423593}$

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:  $11$
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:  \( 58001.1435756 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times D_5$ (as 20T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 16 conjugacy class representatives for $C_2^2\times D_5$
Character table for $C_2^2\times D_5$

Intermediate fields

\(\Q(\sqrt{357}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{17}, \sqrt{21})\), 5.1.14161.1, 10.2.5798839393557.1, 10.2.341108199621.1, 10.2.3409076657.1

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 20 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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
3Data not computed
$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$$[\ ]_{2}^{2}$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$