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Properties

Label 20.0.12675450570...5625.3
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 7^{10}\cdot 11^{16}$
Root discriminant $40.29$
Ramified primes $5, 7, 11$
Class number $275$ (GRH)
Class group $[5, 55]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![512579, -768615, 1330894, -1372952, 1343738, -1005043, 695944, -392889, 203504, -88398, 38496, -16253, 8065, -3815, 1385, -166, -57, 0, 21, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 21*x^18 - 57*x^16 - 166*x^15 + 1385*x^14 - 3815*x^13 + 8065*x^12 - 16253*x^11 + 38496*x^10 - 88398*x^9 + 203504*x^8 - 392889*x^7 + 695944*x^6 - 1005043*x^5 + 1343738*x^4 - 1372952*x^3 + 1330894*x^2 - 768615*x + 512579)
 
gp: K = bnfinit(x^20 - 8*x^19 + 21*x^18 - 57*x^16 - 166*x^15 + 1385*x^14 - 3815*x^13 + 8065*x^12 - 16253*x^11 + 38496*x^10 - 88398*x^9 + 203504*x^8 - 392889*x^7 + 695944*x^6 - 1005043*x^5 + 1343738*x^4 - 1372952*x^3 + 1330894*x^2 - 768615*x + 512579, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 21 x^{18} - 57 x^{16} - 166 x^{15} + 1385 x^{14} - 3815 x^{13} + 8065 x^{12} - 16253 x^{11} + 38496 x^{10} - 88398 x^{9} + 203504 x^{8} - 392889 x^{7} + 695944 x^{6} - 1005043 x^{5} + 1343738 x^{4} - 1372952 x^{3} + 1330894 x^{2} - 768615 x + 512579 \)

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:  \(126754505709914865311944228515625=5^{10}\cdot 7^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.29$
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:  $5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(385=5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{385}(64,·)$, $\chi_{385}(1,·)$, $\chi_{385}(69,·)$, $\chi_{385}(71,·)$, $\chi_{385}(141,·)$, $\chi_{385}(146,·)$, $\chi_{385}(279,·)$, $\chi_{385}(344,·)$, $\chi_{385}(356,·)$, $\chi_{385}(34,·)$, $\chi_{385}(251,·)$, $\chi_{385}(36,·)$, $\chi_{385}(104,·)$, $\chi_{385}(169,·)$, $\chi_{385}(174,·)$, $\chi_{385}(111,·)$, $\chi_{385}(181,·)$, $\chi_{385}(246,·)$, $\chi_{385}(379,·)$, $\chi_{385}(309,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{331} a^{18} + \frac{122}{331} a^{17} + \frac{152}{331} a^{16} + \frac{100}{331} a^{15} + \frac{39}{331} a^{14} - \frac{49}{331} a^{13} - \frac{108}{331} a^{12} - \frac{159}{331} a^{11} + \frac{13}{331} a^{10} - \frac{124}{331} a^{9} - \frac{51}{331} a^{8} + \frac{113}{331} a^{7} - \frac{100}{331} a^{6} + \frac{10}{331} a^{5} + \frac{146}{331} a^{4} - \frac{79}{331} a^{3} - \frac{87}{331} a^{2} - \frac{3}{331} a - \frac{47}{331}$, $\frac{1}{281972049288892662160127749488414700450379724203057} a^{19} + \frac{281213443032968911968641456778440316350377154158}{281972049288892662160127749488414700450379724203057} a^{18} - \frac{34635945451423344263275564734385420256615672551610}{281972049288892662160127749488414700450379724203057} a^{17} + \frac{75291123344253318937575626251681865539510641473915}{281972049288892662160127749488414700450379724203057} a^{16} - \frac{110872290418637179649794640145626914355944918800185}{281972049288892662160127749488414700450379724203057} a^{15} - \frac{131400499085799544136446265647699330448235599022067}{281972049288892662160127749488414700450379724203057} a^{14} - \frac{69506941990299352543029628282726342906929329613842}{281972049288892662160127749488414700450379724203057} a^{13} - \frac{43212787382876313518344220132757987527558556024655}{281972049288892662160127749488414700450379724203057} a^{12} + \frac{139841693591235270656467794736697102416489486817890}{281972049288892662160127749488414700450379724203057} a^{11} - \frac{7124659925890979261353682940108196039468667870251}{281972049288892662160127749488414700450379724203057} a^{10} - \frac{119029735989126447340531284157136178050147598731477}{281972049288892662160127749488414700450379724203057} a^{9} - \frac{136287305793862390924203204940465853206327097369352}{281972049288892662160127749488414700450379724203057} a^{8} - \frac{86821888747271054807698948016493937357967953425220}{281972049288892662160127749488414700450379724203057} a^{7} - \frac{34935331559202675488937988060752103100687678009573}{281972049288892662160127749488414700450379724203057} a^{6} - \frac{15972829145431802251001808662560699854206766182159}{281972049288892662160127749488414700450379724203057} a^{5} - \frac{61931094602540962817276018910262571984157249994682}{281972049288892662160127749488414700450379724203057} a^{4} + \frac{140801700959187768629647234755707769172646190366}{918475730582712254593249998333598372802539818251} a^{3} + \frac{12847755937609852104433400455768716629006219940782}{281972049288892662160127749488414700450379724203057} a^{2} - \frac{132636326164832780707009889589642435564630458927882}{281972049288892662160127749488414700450379724203057} a - \frac{94429279588791837854024744361719452702291514609365}{281972049288892662160127749488414700450379724203057}$

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

Class group and class number

$C_{5}\times C_{55}$, which has order $275$ (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:  \( 140644.599182 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1, 10.0.11258530353021875.4, 10.0.3602729712967.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
7Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$