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Properties

Label 20.4.27460473324...5625.1
Degree $20$
Signature $[4, 8]$
Discriminant $3^{20}\cdot 5^{10}\cdot 73^{8}$
Root discriminant $37.32$
Ramified primes $3, 5, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T230

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![77841, 0, -78570, 0, 23949, 0, -7488, 0, 8784, 0, -5979, 0, 1966, 0, -266, 0, 12, 0, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 + 12*x^16 - 266*x^14 + 1966*x^12 - 5979*x^10 + 8784*x^8 - 7488*x^6 + 23949*x^4 - 78570*x^2 + 77841)
 
gp: K = bnfinit(x^20 - 5*x^18 + 12*x^16 - 266*x^14 + 1966*x^12 - 5979*x^10 + 8784*x^8 - 7488*x^6 + 23949*x^4 - 78570*x^2 + 77841, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{18} + 12 x^{16} - 266 x^{14} + 1966 x^{12} - 5979 x^{10} + 8784 x^{8} - 7488 x^{6} + 23949 x^{4} - 78570 x^{2} + 77841 \)

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:  \(27460473324661212646782041015625=3^{20}\cdot 5^{10}\cdot 73^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.32$
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:  $3, 5, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{9} - \frac{1}{2} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{10} - \frac{1}{2} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{18} a^{14} + \frac{1}{18} a^{12} - \frac{1}{2} a^{9} - \frac{5}{18} a^{8} + \frac{1}{18} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{54} a^{15} + \frac{1}{54} a^{13} + \frac{2}{27} a^{9} - \frac{17}{54} a^{7} - \frac{1}{2} a^{6} - \frac{4}{9} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{54} a^{16} + \frac{1}{54} a^{14} + \frac{2}{27} a^{10} - \frac{17}{54} a^{8} - \frac{1}{2} a^{7} - \frac{4}{9} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{54} a^{17} - \frac{1}{54} a^{13} + \frac{2}{27} a^{11} - \frac{7}{18} a^{9} - \frac{1}{2} a^{8} - \frac{7}{54} a^{7} + \frac{4}{9} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{226376285458756626} a^{18} + \frac{958710743611754}{113188142729378313} a^{16} + \frac{4947573037653}{2794768956280946} a^{14} + \frac{5082894493023865}{226376285458756626} a^{12} + \frac{22457913313686971}{113188142729378313} a^{10} - \frac{1}{2} a^{9} - \frac{33350492668267853}{75458761819585542} a^{8} - \frac{1}{2} a^{7} + \frac{4173684958971805}{25152920606528514} a^{6} + \frac{10401947178605971}{25152920606528514} a^{4} - \frac{1}{2} a^{3} + \frac{822753112504249}{2794768956280946} a^{2} + \frac{688672225938860}{1397384478140473}$, $\frac{1}{7017664849221455406} a^{19} - \frac{19043345684883587}{7017664849221455406} a^{17} - \frac{3426668959342867}{1169610808203575901} a^{15} + \frac{172323661340579402}{3508832424610727703} a^{13} + \frac{55995140789058323}{3508832424610727703} a^{11} + \frac{30137133683571919}{1169610808203575901} a^{9} - \frac{1}{2} a^{8} - \frac{64288920232186565}{389870269401191967} a^{7} + \frac{95332272429363494}{389870269401191967} a^{5} - \frac{1}{2} a^{4} - \frac{103732961001163201}{259913512934127978} a^{3} - \frac{19583422720229375}{86637837644709326} a$

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:  \( 93153413.8798 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T230:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1920
The 24 conjugacy class representatives for t20n230
Character table for t20n230 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.10791225.1, 10.2.5240274165028125.1, 10.10.582252685003125.1, 10.2.1048054833005625.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.12.14.6$x^{12} + 3 x^{11} + 3 x^{10} - 6 x^{9} + 3 x^{8} + 9 x^{7} + 9 x^{4} + 9 x^{3} + 9$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$73$73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.6.4.2$x^{6} - 73 x^{3} + 58619$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
73.6.4.2$x^{6} - 73 x^{3} + 58619$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$