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Properties

Label 20.20.1138643660...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{30}\cdot 5^{10}\cdot 19^{4}\cdot 1699^{4}$
Root discriminant $50.45$
Ramified primes $2, 5, 19, 1699$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times D_5\wr C_2$ (as 20T100)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -48, 666, -1932, -5131, 19142, 11954, -57636, -7386, 70398, -4751, -41050, 6835, 12076, -2585, -1840, 435, 138, -34, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 34*x^18 + 138*x^17 + 435*x^16 - 1840*x^15 - 2585*x^14 + 12076*x^13 + 6835*x^12 - 41050*x^11 - 4751*x^10 + 70398*x^9 - 7386*x^8 - 57636*x^7 + 11954*x^6 + 19142*x^5 - 5131*x^4 - 1932*x^3 + 666*x^2 - 48*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 - 34*x^18 + 138*x^17 + 435*x^16 - 1840*x^15 - 2585*x^14 + 12076*x^13 + 6835*x^12 - 41050*x^11 - 4751*x^10 + 70398*x^9 - 7386*x^8 - 57636*x^7 + 11954*x^6 + 19142*x^5 - 5131*x^4 - 1932*x^3 + 666*x^2 - 48*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 34 x^{18} + 138 x^{17} + 435 x^{16} - 1840 x^{15} - 2585 x^{14} + 12076 x^{13} + 6835 x^{12} - 41050 x^{11} - 4751 x^{10} + 70398 x^{9} - 7386 x^{8} - 57636 x^{7} + 11954 x^{6} + 19142 x^{5} - 5131 x^{4} - 1932 x^{3} + 666 x^{2} - 48 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11386436606367760604200960000000000=2^{30}\cdot 5^{10}\cdot 19^{4}\cdot 1699^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.45$
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:  $2, 5, 19, 1699$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{17882394677449659579002170357} a^{19} - \frac{4569637586165575957043337713}{17882394677449659579002170357} a^{18} - \frac{830227370107053980738051192}{17882394677449659579002170357} a^{17} - \frac{8213240978215623419930449155}{17882394677449659579002170357} a^{16} + \frac{2069160530646722664439298207}{17882394677449659579002170357} a^{15} + \frac{4429066484734937455702512390}{17882394677449659579002170357} a^{14} + \frac{1438179280324719272448024866}{17882394677449659579002170357} a^{13} - \frac{6531142659577013965193779876}{17882394677449659579002170357} a^{12} + \frac{4856766945759148141268770477}{17882394677449659579002170357} a^{11} - \frac{4009932175595294540975574593}{17882394677449659579002170357} a^{10} + \frac{1670193847072895716146408608}{17882394677449659579002170357} a^{9} - \frac{3125405350698286898951576646}{17882394677449659579002170357} a^{8} - \frac{3742384072456447478723179824}{17882394677449659579002170357} a^{7} - \frac{4587489911490165512849603767}{17882394677449659579002170357} a^{6} - \frac{2650258689595998812023565261}{17882394677449659579002170357} a^{5} + \frac{5354925492988961684419228506}{17882394677449659579002170357} a^{4} - \frac{8886854071060826651264329924}{17882394677449659579002170357} a^{3} + \frac{940276557014718546792750744}{17882394677449659579002170357} a^{2} + \frac{218131053972032819288783706}{1625672243404514507182015487} a - \frac{1267311173795352016054903227}{17882394677449659579002170357}$

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:  $19$
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:  \( 29084821168.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_5\wr C_2$ (as 20T100):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $C_2\times D_5\wr C_2$
Character table for $C_2\times D_5\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 10.10.3256446753125.1

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

Sibling fields

Degree 20 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
2Data not computed
5Data not computed
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
1699Data not computed