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Properties

Label 20.4.12889899207...3296.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{38}\cdot 97^{2}\cdot 2657^{4}$
Root discriminant $28.54$
Ramified primes $2, 97, 2657$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1028

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-32, 0, 192, -704, 1520, -1848, 1825, -1234, 775, -1140, 1134, -1256, 892, -388, 286, -108, 82, -8, 15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 15*x^18 - 8*x^17 + 82*x^16 - 108*x^15 + 286*x^14 - 388*x^13 + 892*x^12 - 1256*x^11 + 1134*x^10 - 1140*x^9 + 775*x^8 - 1234*x^7 + 1825*x^6 - 1848*x^5 + 1520*x^4 - 704*x^3 + 192*x^2 - 32)
 
gp: K = bnfinit(x^20 - 2*x^19 + 15*x^18 - 8*x^17 + 82*x^16 - 108*x^15 + 286*x^14 - 388*x^13 + 892*x^12 - 1256*x^11 + 1134*x^10 - 1140*x^9 + 775*x^8 - 1234*x^7 + 1825*x^6 - 1848*x^5 + 1520*x^4 - 704*x^3 + 192*x^2 - 32, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 15 x^{18} - 8 x^{17} + 82 x^{16} - 108 x^{15} + 286 x^{14} - 388 x^{13} + 892 x^{12} - 1256 x^{11} + 1134 x^{10} - 1140 x^{9} + 775 x^{8} - 1234 x^{7} + 1825 x^{6} - 1848 x^{5} + 1520 x^{4} - 704 x^{3} + 192 x^{2} - 32 \)

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:  \(128898992075490807544645943296=2^{38}\cdot 97^{2}\cdot 2657^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.54$
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:  $2, 97, 2657$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{16} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{2} a^{10} - \frac{1}{4} a^{8} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2167096502528011085115778654288} a^{19} - \frac{4316247264560405245710343811}{541774125632002771278944663572} a^{18} + \frac{3569920091004685131166086871}{2167096502528011085115778654288} a^{17} + \frac{265974082858570608716231048337}{1083548251264005542557889327144} a^{16} - \frac{517330290323075394775105165921}{1083548251264005542557889327144} a^{15} - \frac{35489692500314024928713328121}{270887062816001385639472331786} a^{14} - \frac{199994505558889605703180901345}{1083548251264005542557889327144} a^{13} + \frac{17963132699774909132381898425}{135443531408000692819736165893} a^{12} - \frac{216995312231330177415150631265}{541774125632002771278944663572} a^{11} - \frac{39173052000623953483476813424}{135443531408000692819736165893} a^{10} + \frac{493894863452848761410731933575}{1083548251264005542557889327144} a^{9} - \frac{5568859544864499526194800707}{135443531408000692819736165893} a^{8} + \frac{1062371067065988205665553955127}{2167096502528011085115778654288} a^{7} + \frac{121285127978761780030977279079}{270887062816001385639472331786} a^{6} - \frac{329655696049927623480400496175}{2167096502528011085115778654288} a^{5} - \frac{115853772293203131587160381417}{1083548251264005542557889327144} a^{4} - \frac{77892841517041692706386578891}{541774125632002771278944663572} a^{3} + \frac{10923669870672365804217335751}{270887062816001385639472331786} a^{2} + \frac{34833046195505737392985692251}{135443531408000692819736165893} a - \frac{11327852867807984570737253451}{135443531408000692819736165893}$

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:  \( 11745334.7428 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1028:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 228 conjugacy class representatives for t20n1028 are not computed
Character table for t20n1028 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.6.925322313728.1

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

Sibling fields

Degree 20 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
97Data not computed
2657Data not computed