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Properties

Label 20.0.92463633619...5625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 79^{10}$
Root discriminant $19.87$
Ramified primes $5, 79$
Class number $4$
Class group $[4]$
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![441, -1407, 3907, -10840, 20704, -26479, 24262, -17489, 10155, -4142, 1464, -926, 667, -374, 230, -142, 72, -34, 17, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 17*x^18 - 34*x^17 + 72*x^16 - 142*x^15 + 230*x^14 - 374*x^13 + 667*x^12 - 926*x^11 + 1464*x^10 - 4142*x^9 + 10155*x^8 - 17489*x^7 + 24262*x^6 - 26479*x^5 + 20704*x^4 - 10840*x^3 + 3907*x^2 - 1407*x + 441)
 
gp: K = bnfinit(x^20 - 6*x^19 + 17*x^18 - 34*x^17 + 72*x^16 - 142*x^15 + 230*x^14 - 374*x^13 + 667*x^12 - 926*x^11 + 1464*x^10 - 4142*x^9 + 10155*x^8 - 17489*x^7 + 24262*x^6 - 26479*x^5 + 20704*x^4 - 10840*x^3 + 3907*x^2 - 1407*x + 441, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 17 x^{18} - 34 x^{17} + 72 x^{16} - 142 x^{15} + 230 x^{14} - 374 x^{13} + 667 x^{12} - 926 x^{11} + 1464 x^{10} - 4142 x^{9} + 10155 x^{8} - 17489 x^{7} + 24262 x^{6} - 26479 x^{5} + 20704 x^{4} - 10840 x^{3} + 3907 x^{2} - 1407 x + 441 \)

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:  \(92463633619402804697265625=5^{10}\cdot 79^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.87$
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:  $5, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{4}{9} a^{4} + \frac{2}{9} a^{2} + \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{4}{9} a^{5} + \frac{2}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{2}{9} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{2}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{3} a^{8} - \frac{1}{9} a^{7} - \frac{1}{3} a^{4} + \frac{1}{9} a^{3} - \frac{4}{9} a^{2} - \frac{2}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{11} + \frac{1}{9} a^{9} - \frac{4}{9} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{4}{9} a^{3} + \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{261} a^{17} - \frac{10}{261} a^{16} - \frac{2}{87} a^{14} + \frac{2}{87} a^{13} + \frac{2}{261} a^{12} + \frac{1}{9} a^{11} - \frac{28}{261} a^{10} + \frac{14}{261} a^{9} + \frac{121}{261} a^{8} + \frac{25}{87} a^{7} + \frac{14}{29} a^{6} - \frac{20}{87} a^{5} - \frac{5}{261} a^{4} + \frac{10}{261} a^{3} - \frac{65}{261} a^{2} + \frac{26}{87} a + \frac{1}{29}$, $\frac{1}{22082949} a^{18} + \frac{8128}{7360983} a^{17} - \frac{304420}{7360983} a^{16} + \frac{1163300}{22082949} a^{15} - \frac{346384}{7360983} a^{14} + \frac{300385}{22082949} a^{13} + \frac{559217}{22082949} a^{12} - \frac{1452319}{22082949} a^{11} - \frac{10765}{1051569} a^{10} + \frac{592988}{7360983} a^{9} - \frac{187549}{432999} a^{8} - \frac{9783113}{22082949} a^{7} + \frac{2766413}{7360983} a^{6} + \frac{925433}{22082949} a^{5} + \frac{10462315}{22082949} a^{4} + \frac{1267372}{22082949} a^{3} - \frac{11854}{39933} a^{2} + \frac{3649087}{7360983} a + \frac{114530}{350523}$, $\frac{1}{45657458656668529943013} a^{19} - \frac{23195417643343}{45657458656668529943013} a^{18} - \frac{6050933711007376787}{15219152885556176647671} a^{17} - \frac{1944489702227716068589}{45657458656668529943013} a^{16} - \frac{777936617014386544583}{45657458656668529943013} a^{15} - \frac{1324173157488029812388}{45657458656668529943013} a^{14} - \frac{1904478989419317426671}{45657458656668529943013} a^{13} + \frac{120696819997983601102}{15219152885556176647671} a^{12} + \frac{1872918758942328451120}{45657458656668529943013} a^{11} - \frac{765344633949518229307}{5073050961852058882557} a^{10} + \frac{2223683714799178800095}{15219152885556176647671} a^{9} - \frac{15890199285860672837273}{45657458656668529943013} a^{8} + \frac{10944838650458319780977}{45657458656668529943013} a^{7} - \frac{13411006408559846620510}{45657458656668529943013} a^{6} - \frac{21476332304833541376412}{45657458656668529943013} a^{5} + \frac{3977642261898796570801}{15219152885556176647671} a^{4} - \frac{2236201348964008551668}{45657458656668529943013} a^{3} + \frac{10202945659093834493437}{45657458656668529943013} a^{2} + \frac{609903413998536700346}{15219152885556176647671} a + \frac{259454546615193572719}{724721565978865554651}$

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

Class group and class number

$C_{4}$, which has order $4$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 24221.8358828 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-79}) \), \(\Q(\sqrt{-395}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-79})\), 5.1.6241.1 x5, 10.0.3077056399.1, 10.0.9615801246875.1 x5, 10.2.121719003125.1 x5

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$79$79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$