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Properties

Label 20.20.5233381311...6896.2
Degree $20$
Signature $[20, 0]$
Discriminant $2^{30}\cdot 3^{6}\cdot 401^{8}$
Root discriminant $43.25$
Ramified primes $2, 3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:D_5$ (as 20T85)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -42, -46, 1070, 989, -7620, -8578, 20366, 29257, -16388, -34284, 3110, 18403, 1598, -4924, -766, 648, 102, -40, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 40*x^18 + 102*x^17 + 648*x^16 - 766*x^15 - 4924*x^14 + 1598*x^13 + 18403*x^12 + 3110*x^11 - 34284*x^10 - 16388*x^9 + 29257*x^8 + 20366*x^7 - 8578*x^6 - 7620*x^5 + 989*x^4 + 1070*x^3 - 46*x^2 - 42*x - 1)
 
gp: K = bnfinit(x^20 - 4*x^19 - 40*x^18 + 102*x^17 + 648*x^16 - 766*x^15 - 4924*x^14 + 1598*x^13 + 18403*x^12 + 3110*x^11 - 34284*x^10 - 16388*x^9 + 29257*x^8 + 20366*x^7 - 8578*x^6 - 7620*x^5 + 989*x^4 + 1070*x^3 - 46*x^2 - 42*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 40 x^{18} + 102 x^{17} + 648 x^{16} - 766 x^{15} - 4924 x^{14} + 1598 x^{13} + 18403 x^{12} + 3110 x^{11} - 34284 x^{10} - 16388 x^{9} + 29257 x^{8} + 20366 x^{7} - 8578 x^{6} - 7620 x^{5} + 989 x^{4} + 1070 x^{3} - 46 x^{2} - 42 x - 1 \)

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:  \(523338131151796421636247986896896=2^{30}\cdot 3^{6}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.25$
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:  $2, 3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1595622376865644035925747} a^{19} + \frac{180074302223193427673733}{1595622376865644035925747} a^{18} + \frac{242930023009577548192753}{1595622376865644035925747} a^{17} - \frac{722799469874702029406687}{1595622376865644035925747} a^{16} + \frac{12159328920548001096808}{1595622376865644035925747} a^{15} - \frac{183636556091069554195512}{1595622376865644035925747} a^{14} - \frac{692766014947078640369911}{1595622376865644035925747} a^{13} - \frac{154520450398492930762685}{1595622376865644035925747} a^{12} - \frac{310929574599003285177179}{1595622376865644035925747} a^{11} + \frac{598315757736815850279010}{1595622376865644035925747} a^{10} - \frac{684482616465944357174093}{1595622376865644035925747} a^{9} + \frac{126328038437480608357490}{1595622376865644035925747} a^{8} - \frac{2204253479212741433099}{1595622376865644035925747} a^{7} - \frac{14765362049783257228917}{83980125098191791364513} a^{6} - \frac{380767895274268839721500}{1595622376865644035925747} a^{5} + \frac{599961468561717927774787}{1595622376865644035925747} a^{4} + \frac{377192536056324863682890}{1595622376865644035925747} a^{3} - \frac{550912059125601483457029}{1595622376865644035925747} a^{2} + \frac{469202797341102172831250}{1595622376865644035925747} a + \frac{463137675676976695239739}{1595622376865644035925747}$

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:  \( 8654299680.25 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T85):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

5.5.160801.1, 10.10.714893274344448.1, 10.10.79432586038272.1, 10.10.238297758114816.2

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 32 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 R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
401Data not computed