↑

Properties

Label 20.12.1583112947...5449.1
Degree $20$
Signature $[12, 4]$
Discriminant $3^{10}\cdot 401^{9}$
Root discriminant $25.70$
Ramified primes $3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:D_5$ (as 20T87)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -8, -26, -30, 14, 99, 110, 36, -202, -195, 202, 36, -110, 99, -14, -30, 26, -8, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 8*x^18 + 26*x^17 - 30*x^16 - 14*x^15 + 99*x^14 - 110*x^13 + 36*x^12 + 202*x^11 - 195*x^10 - 202*x^9 + 36*x^8 + 110*x^7 + 99*x^6 + 14*x^5 - 30*x^4 - 26*x^3 - 8*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 8*x^18 + 26*x^17 - 30*x^16 - 14*x^15 + 99*x^14 - 110*x^13 + 36*x^12 + 202*x^11 - 195*x^10 - 202*x^9 + 36*x^8 + 110*x^7 + 99*x^6 + 14*x^5 - 30*x^4 - 26*x^3 - 8*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 8 x^{18} + 26 x^{17} - 30 x^{16} - 14 x^{15} + 99 x^{14} - 110 x^{13} + 36 x^{12} + 202 x^{11} - 195 x^{10} - 202 x^{9} + 36 x^{8} + 110 x^{7} + 99 x^{6} + 14 x^{5} - 30 x^{4} - 26 x^{3} - 8 x^{2} + 2 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15831129474510904002158875449=3^{10}\cdot 401^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.70$
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:  $3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{11} + \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{6} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{15} + \frac{1}{6} a^{10} - \frac{1}{2} a^{9} + \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3}$, $\frac{1}{18} a^{16} + \frac{1}{18} a^{15} - \frac{1}{18} a^{14} + \frac{1}{18} a^{12} - \frac{1}{6} a^{11} - \frac{2}{9} a^{10} - \frac{4}{9} a^{9} - \frac{5}{18} a^{8} + \frac{5}{18} a^{7} - \frac{7}{18} a^{6} + \frac{1}{6} a^{5} + \frac{2}{9} a^{4} + \frac{1}{6} a^{3} - \frac{2}{9} a^{2} - \frac{7}{18} a - \frac{4}{9}$, $\frac{1}{18} a^{17} + \frac{1}{18} a^{15} + \frac{1}{18} a^{14} + \frac{1}{18} a^{13} - \frac{1}{18} a^{12} + \frac{1}{9} a^{11} - \frac{1}{18} a^{10} - \frac{1}{6} a^{9} + \frac{2}{9} a^{8} - \frac{1}{3} a^{7} + \frac{1}{18} a^{6} + \frac{2}{9} a^{5} + \frac{1}{9} a^{4} - \frac{7}{18} a^{3} - \frac{1}{2} a^{2} - \frac{2}{9} a - \frac{2}{9}$, $\frac{1}{1564218} a^{18} + \frac{20651}{782109} a^{17} + \frac{7034}{260703} a^{16} - \frac{57292}{782109} a^{15} - \frac{5939}{86901} a^{14} - \frac{1129}{173802} a^{13} + \frac{36295}{521406} a^{12} + \frac{75217}{1564218} a^{11} + \frac{12373}{260703} a^{10} - \frac{89548}{260703} a^{9} - \frac{70307}{260703} a^{8} - \frac{388255}{1564218} a^{7} + \frac{6030}{28967} a^{6} - \frac{16177}{260703} a^{5} - \frac{40117}{260703} a^{4} - \frac{664957}{1564218} a^{3} + \frac{79867}{260703} a^{2} - \frac{124184}{782109} a - \frac{57935}{1564218}$, $\frac{1}{1564218} a^{19} - \frac{33271}{1564218} a^{17} + \frac{4934}{782109} a^{16} + \frac{73709}{1564218} a^{15} + \frac{12856}{260703} a^{14} + \frac{15559}{521406} a^{13} - \frac{26602}{782109} a^{12} + \frac{259355}{1564218} a^{11} + \frac{93943}{521406} a^{10} + \frac{14093}{521406} a^{9} + \frac{443747}{1564218} a^{8} - \frac{763811}{1564218} a^{7} - \frac{24587}{57934} a^{6} - \frac{24001}{173802} a^{5} - \frac{202325}{782109} a^{4} + \frac{775681}{1564218} a^{3} + \frac{409289}{1564218} a^{2} - \frac{8771}{782109} a + \frac{12335}{1564218}$

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:  $15$
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:  \( 7444020.83455 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T87):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

5.5.160801.1, 10.6.2094413889681.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
401Data not computed