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Properties

Label 20.0.21363681729...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 13^{10}$
Root discriminant $52.06$
Ramified primes $2, 3, 5, 13$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^2\times F_5$ (as 20T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![452761, 951210, 689231, 229440, 557193, 319290, -398664, -363852, 139785, 122754, -51423, -20064, 19293, 216, -3762, 156, 453, -12, -31, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 31*x^18 - 12*x^17 + 453*x^16 + 156*x^15 - 3762*x^14 + 216*x^13 + 19293*x^12 - 20064*x^11 - 51423*x^10 + 122754*x^9 + 139785*x^8 - 363852*x^7 - 398664*x^6 + 319290*x^5 + 557193*x^4 + 229440*x^3 + 689231*x^2 + 951210*x + 452761)
 
gp: K = bnfinit(x^20 - 31*x^18 - 12*x^17 + 453*x^16 + 156*x^15 - 3762*x^14 + 216*x^13 + 19293*x^12 - 20064*x^11 - 51423*x^10 + 122754*x^9 + 139785*x^8 - 363852*x^7 - 398664*x^6 + 319290*x^5 + 557193*x^4 + 229440*x^3 + 689231*x^2 + 951210*x + 452761, 1)
 

Normalized defining polynomial

\( x^{20} - 31 x^{18} - 12 x^{17} + 453 x^{16} + 156 x^{15} - 3762 x^{14} + 216 x^{13} + 19293 x^{12} - 20064 x^{11} - 51423 x^{10} + 122754 x^{9} + 139785 x^{8} - 363852 x^{7} - 398664 x^{6} + 319290 x^{5} + 557193 x^{4} + 229440 x^{3} + 689231 x^{2} + 951210 x + 452761 \)

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:  \(21363681729976837664400000000000000=2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.06$
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:  $2, 3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{4} - \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{18} a^{14} - \frac{1}{18} a^{12} + \frac{1}{18} a^{8} + \frac{4}{9} a^{6} - \frac{1}{2} a^{4} + \frac{7}{18} a^{2} + \frac{1}{9}$, $\frac{1}{18} a^{15} - \frac{1}{18} a^{13} + \frac{1}{18} a^{9} + \frac{4}{9} a^{7} - \frac{1}{2} a^{5} + \frac{7}{18} a^{3} + \frac{1}{9} a$, $\frac{1}{180} a^{16} + \frac{1}{45} a^{15} + \frac{1}{45} a^{14} + \frac{7}{90} a^{13} + \frac{1}{18} a^{12} - \frac{1}{30} a^{11} - \frac{7}{90} a^{10} + \frac{11}{90} a^{9} + \frac{1}{180} a^{8} + \frac{19}{90} a^{7} - \frac{4}{9} a^{6} - \frac{1}{15} a^{5} - \frac{8}{45} a^{4} + \frac{16}{45} a^{3} - \frac{7}{90} a^{2} + \frac{37}{90} a + \frac{61}{180}$, $\frac{1}{180} a^{17} - \frac{1}{90} a^{15} - \frac{1}{90} a^{14} + \frac{1}{45} a^{13} + \frac{7}{90} a^{12} + \frac{1}{18} a^{11} - \frac{1}{15} a^{10} + \frac{13}{180} a^{9} + \frac{1}{45} a^{8} + \frac{22}{45} a^{7} + \frac{19}{90} a^{6} + \frac{4}{45} a^{5} - \frac{13}{30} a^{4} - \frac{1}{9} a^{3} - \frac{1}{9} a^{2} - \frac{13}{36} a - \frac{17}{90}$, $\frac{1}{180} a^{18} - \frac{1}{45} a^{15} + \frac{1}{90} a^{14} - \frac{2}{45} a^{13} + \frac{1}{18} a^{12} + \frac{1}{30} a^{11} - \frac{1}{12} a^{10} + \frac{2}{45} a^{9} + \frac{1}{9} a^{8} + \frac{16}{45} a^{7} + \frac{19}{45} a^{6} - \frac{7}{30} a^{5} + \frac{1}{30} a^{4} + \frac{17}{45} a^{3} + \frac{77}{180} a^{2} - \frac{14}{45} a + \frac{1}{15}$, $\frac{1}{229114094078754013199906577154443200577095456340} a^{19} + \frac{198728261194832979850818437947316523586490257}{114557047039377006599953288577221600288547728170} a^{18} - \frac{5172469372979306860589292427467952026864635}{11455704703937700659995328857722160028854772817} a^{17} + \frac{50177422255026539501979041580907420233995743}{57278523519688503299976644288610800144273864085} a^{16} - \frac{199729698693384389926054410360040331725386964}{11455704703937700659995328857722160028854772817} a^{15} + \frac{65236580538556509591880305213835797100402439}{38185682346459002199984429525740533429515909390} a^{14} - \frac{2341313023288536620583305274341090796332051093}{38185682346459002199984429525740533429515909390} a^{13} - \frac{1927281465416071071726652038387216966790098797}{114557047039377006599953288577221600288547728170} a^{12} - \frac{5151875650507330310630185406327093768391539917}{229114094078754013199906577154443200577095456340} a^{11} + \frac{1355617173975270113110278534470088834106678607}{114557047039377006599953288577221600288547728170} a^{10} + \frac{7395136693443408485195029433466133513815892036}{57278523519688503299976644288610800144273864085} a^{9} - \frac{7735725652606178496290822778610026971074108}{1272856078215300073332814317524684447650530313} a^{8} + \frac{1716412371237636859427470171039312422822165589}{7637136469291800439996885905148106685903181878} a^{7} - \frac{40591298329671992700618931645262538858458568449}{114557047039377006599953288577221600288547728170} a^{6} + \frac{43994501228822297786216240598199172338468258153}{114557047039377006599953288577221600288547728170} a^{5} - \frac{1298049075946048230923779195275579896131134292}{11455704703937700659995328857722160028854772817} a^{4} - \frac{44113312232950166651500787802417385119413755727}{229114094078754013199906577154443200577095456340} a^{3} - \frac{5965625380355943283361370741328560577800840421}{19092841173229501099992214762870266714757954695} a^{2} + \frac{23034977196843204327048156055574015134193512733}{57278523519688503299976644288610800144273864085} a - \frac{249499539897598711697013072200720822744541586}{6364280391076500366664071587623422238252651565}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 756002700.5372794 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times F_5$ (as 20T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_2^2\times F_5$
Character table for $C_2^2\times F_5$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-195}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{13}, \sqrt{-15})\), 5.1.162000.1, 10.0.393660000000.1, 10.0.146163202380000000.1, 10.2.9744213492000000.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$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}$
$3$3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$