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Properties

Label 20.4.29524500000...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{30}\cdot 3^{10}\cdot 5^{31}$
Root discriminant $59.36$
Ramified primes $2, 3, 5$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4837294080, 0, -1007769600, 0, -293932800, 0, 83980800, 0, 2332800, 0, -2099520, 0, 174960, 0, -6480, 0, 540, 0, -30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 30*x^18 + 540*x^16 - 6480*x^14 + 174960*x^12 - 2099520*x^10 + 2332800*x^8 + 83980800*x^6 - 293932800*x^4 - 1007769600*x^2 + 4837294080)
 
gp: K = bnfinit(x^20 - 30*x^18 + 540*x^16 - 6480*x^14 + 174960*x^12 - 2099520*x^10 + 2332800*x^8 + 83980800*x^6 - 293932800*x^4 - 1007769600*x^2 + 4837294080, 1)
 

Normalized defining polynomial

\( x^{20} - 30 x^{18} + 540 x^{16} - 6480 x^{14} + 174960 x^{12} - 2099520 x^{10} + 2332800 x^{8} + 83980800 x^{6} - 293932800 x^{4} - 1007769600 x^{2} + 4837294080 \)

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:  \(295245000000000000000000000000000000=2^{30}\cdot 3^{10}\cdot 5^{31}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.36$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $\frac{1}{6} a^{2}$, $\frac{1}{6} a^{3}$, $\frac{1}{36} a^{4}$, $\frac{1}{36} a^{5}$, $\frac{1}{216} a^{6}$, $\frac{1}{216} a^{7}$, $\frac{1}{1296} a^{8}$, $\frac{1}{1296} a^{9}$, $\frac{1}{7776} a^{10}$, $\frac{1}{7776} a^{11}$, $\frac{1}{46656} a^{12}$, $\frac{1}{46656} a^{13}$, $\frac{1}{279936} a^{14}$, $\frac{1}{279936} a^{15}$, $\frac{1}{5038848} a^{16} + \frac{1}{139968} a^{12} - \frac{1}{23328} a^{10} - \frac{1}{3888} a^{8} + \frac{1}{108} a^{4} - \frac{1}{18} a^{2} + \frac{1}{3}$, $\frac{1}{10077696} a^{17} - \frac{1}{559872} a^{15} + \frac{1}{279936} a^{13} + \frac{1}{23328} a^{11} - \frac{1}{7776} a^{9} - \frac{1}{108} a^{5} + \frac{1}{18} a^{3} + \frac{1}{6} a$, $\frac{1}{215851517650372608} a^{18} - \frac{2036262541}{35975252941728768} a^{16} + \frac{2475704179}{5995875490288128} a^{14} - \frac{3331366495}{499656290857344} a^{12} + \frac{650971937}{55517365650816} a^{10} + \frac{2800690631}{13879341412704} a^{8} - \frac{1484233081}{2313223568784} a^{6} + \frac{1605116785}{192768630732} a^{4} + \frac{6185082881}{128512420488} a^{2} - \frac{717859552}{5354684187}$, $\frac{1}{431703035300745216} a^{19} - \frac{2036262541}{71950505883457536} a^{17} - \frac{18943032569}{11991750980576256} a^{15} - \frac{3331366495}{999312581714688} a^{13} - \frac{80106259}{1370799151872} a^{11} - \frac{7908677743}{27758682825408} a^{9} + \frac{9225135293}{4626447137568} a^{7} + \frac{1605116785}{385537261464} a^{5} + \frac{6185082881}{257024840976} a^{3} + \frac{4636824635}{10709368374} a$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.72000.1, 5.1.78125.1, 10.2.30517578125.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$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}$
5Data not computed