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Properties

Label 20.20.4115394215...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{28}\cdot 5^{15}\cdot 3469^{5}$
Root discriminant $67.72$
Ramified primes $2, 5, 3469$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_5^2:C_4$ (as 20T94)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-124, 16872, 133648, -469792, -798688, 2274052, 582960, -3209444, 353024, 1945052, -486202, -566144, 169380, 84064, -25582, -6574, 1914, 258, -70, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 70*x^18 + 258*x^17 + 1914*x^16 - 6574*x^15 - 25582*x^14 + 84064*x^13 + 169380*x^12 - 566144*x^11 - 486202*x^10 + 1945052*x^9 + 353024*x^8 - 3209444*x^7 + 582960*x^6 + 2274052*x^5 - 798688*x^4 - 469792*x^3 + 133648*x^2 + 16872*x - 124)
 
gp: K = bnfinit(x^20 - 4*x^19 - 70*x^18 + 258*x^17 + 1914*x^16 - 6574*x^15 - 25582*x^14 + 84064*x^13 + 169380*x^12 - 566144*x^11 - 486202*x^10 + 1945052*x^9 + 353024*x^8 - 3209444*x^7 + 582960*x^6 + 2274052*x^5 - 798688*x^4 - 469792*x^3 + 133648*x^2 + 16872*x - 124, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 70 x^{18} + 258 x^{17} + 1914 x^{16} - 6574 x^{15} - 25582 x^{14} + 84064 x^{13} + 169380 x^{12} - 566144 x^{11} - 486202 x^{10} + 1945052 x^{9} + 353024 x^{8} - 3209444 x^{7} + 582960 x^{6} + 2274052 x^{5} - 798688 x^{4} - 469792 x^{3} + 133648 x^{2} + 16872 x - 124 \)

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:  \(4115394215817120555008000000000000000=2^{28}\cdot 5^{15}\cdot 3469^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.72$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{2} a^{16}$, $\frac{1}{2} a^{17}$, $\frac{1}{2} a^{18}$, $\frac{1}{209461642619774446284958265448115270670319262} a^{19} - \frac{9517979354722555772276799381639401867053075}{104730821309887223142479132724057635335159631} a^{18} - \frac{16751593080519109979738295486521753878866998}{104730821309887223142479132724057635335159631} a^{17} + \frac{778397513521090524146931543982337966357631}{209461642619774446284958265448115270670319262} a^{16} - \frac{45113697401751332982682615292761411638254769}{209461642619774446284958265448115270670319262} a^{15} - \frac{43676092732613593749628121412705481952671487}{209461642619774446284958265448115270670319262} a^{14} - \frac{4533072288410355570987751424343631209734953}{104730821309887223142479132724057635335159631} a^{13} + \frac{481390747351669123568983868053190451355566}{104730821309887223142479132724057635335159631} a^{12} - \frac{31640029638704411842721259662017393472863873}{209461642619774446284958265448115270670319262} a^{11} + \frac{38885428048019395247661486532600071307725373}{209461642619774446284958265448115270670319262} a^{10} + \frac{40706500424032102818421733662138100199329934}{104730821309887223142479132724057635335159631} a^{9} - \frac{44393869692789351427034118964299765962446041}{104730821309887223142479132724057635335159631} a^{8} + \frac{32043407959340599361465367829800172685340332}{104730821309887223142479132724057635335159631} a^{7} + \frac{2058995135009927809284981431080238564552728}{104730821309887223142479132724057635335159631} a^{6} - \frac{48350227708313651664555087115408919906521596}{104730821309887223142479132724057635335159631} a^{5} - \frac{9150273115506894507989092639547556359721167}{104730821309887223142479132724057635335159631} a^{4} + \frac{15780346140888567772563108867100729532481049}{104730821309887223142479132724057635335159631} a^{3} - \frac{47429679054726442691278532043960200366825077}{104730821309887223142479132724057635335159631} a^{2} - \frac{16353611211394080053992438683368568160770032}{104730821309887223142479132724057635335159631} a + \frac{437133899738254745811354527723401138395067}{3378413590641523327176746216905085010811601}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 550729563654 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5^2:C_4$ (as 20T94):

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 $D_5^2:C_4$
Character table for $D_5^2:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.6938000.1, 10.10.9627168800000.1

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 $20$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
3469Data not computed