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Properties

Label 20.0.99150356443...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 3^{8}\cdot 5^{10}\cdot 7^{8}$
Root discriminant $19.94$
Ramified primes $2, 3, 5, 7$
Class number $1$
Class group Trivial
Galois group $D_5\wr C_2$ (as 20T55)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20, -120, 484, -1392, 3184, -6036, 9760, -13652, 16717, -18004, 17101, -14310, 10523, -6756, 3754, -1780, 707, -228, 57, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 57*x^18 - 228*x^17 + 707*x^16 - 1780*x^15 + 3754*x^14 - 6756*x^13 + 10523*x^12 - 14310*x^11 + 17101*x^10 - 18004*x^9 + 16717*x^8 - 13652*x^7 + 9760*x^6 - 6036*x^5 + 3184*x^4 - 1392*x^3 + 484*x^2 - 120*x + 20)
 
gp: K = bnfinit(x^20 - 10*x^19 + 57*x^18 - 228*x^17 + 707*x^16 - 1780*x^15 + 3754*x^14 - 6756*x^13 + 10523*x^12 - 14310*x^11 + 17101*x^10 - 18004*x^9 + 16717*x^8 - 13652*x^7 + 9760*x^6 - 6036*x^5 + 3184*x^4 - 1392*x^3 + 484*x^2 - 120*x + 20, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 57 x^{18} - 228 x^{17} + 707 x^{16} - 1780 x^{15} + 3754 x^{14} - 6756 x^{13} + 10523 x^{12} - 14310 x^{11} + 17101 x^{10} - 18004 x^{9} + 16717 x^{8} - 13652 x^{7} + 9760 x^{6} - 6036 x^{5} + 3184 x^{4} - 1392 x^{3} + 484 x^{2} - 120 x + 20 \)

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:  \(99150356443299840000000000=2^{28}\cdot 3^{8}\cdot 5^{10}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.94$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{13} + \frac{1}{4} a^{11} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{17} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{17} - \frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{3}{8} a^{11} - \frac{1}{2} a^{10} + \frac{3}{8} a^{8} + \frac{3}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2}$, $\frac{1}{18312} a^{19} + \frac{1135}{18312} a^{18} - \frac{65}{2289} a^{17} - \frac{61}{2289} a^{16} + \frac{463}{18312} a^{15} + \frac{1885}{18312} a^{14} + \frac{421}{6104} a^{13} + \frac{629}{6104} a^{12} - \frac{853}{4578} a^{11} - \frac{245}{654} a^{10} - \frac{221}{872} a^{9} + \frac{203}{2616} a^{8} - \frac{2159}{9156} a^{7} - \frac{2183}{9156} a^{6} - \frac{967}{4578} a^{5} + \frac{593}{9156} a^{4} + \frac{62}{763} a^{3} + \frac{709}{1526} a^{2} - \frac{2249}{4578} a - \frac{5}{2289}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{607}{2289} a^{19} - \frac{11533}{4578} a^{18} + \frac{62287}{4578} a^{17} - \frac{117674}{2289} a^{16} + \frac{342844}{2289} a^{15} - \frac{806033}{2289} a^{14} + \frac{524886}{763} a^{13} - \frac{1733693}{1526} a^{12} + \frac{3674306}{2289} a^{11} - \frac{1277683}{654} a^{10} + \frac{223918}{109} a^{9} - \frac{609259}{327} a^{8} + \frac{3316640}{2289} a^{7} - \frac{2187784}{2289} a^{6} + \frac{2388703}{4578} a^{5} - \frac{1030027}{4578} a^{4} + \frac{52334}{763} a^{3} - \frac{8269}{763} a^{2} - \frac{5885}{2289} a + \frac{3188}{2289} \) (order $4$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 319835.417345 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\wr C_2$ (as 20T55):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 14 conjugacy class representatives for $D_5\wr C_2$
Character table for $D_5\wr C_2$

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{5})\), 10.0.9957427200000.1

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 25 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 R R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.10.8.1$x^{10} - 7 x^{5} + 147$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$