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Properties

Label 20.0.35228460210...4768.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{15}\cdot 401^{10}$
Root discriminant $33.68$
Ramified primes $2, 401$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $C_5:D_4$ (as 20T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7907, 3320, -20666, 7050, 8749, -17644, 17188, -8380, 7725, -3928, 3542, -1854, 1043, -666, 288, -150, 60, -12, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 8*x^18 - 12*x^17 + 60*x^16 - 150*x^15 + 288*x^14 - 666*x^13 + 1043*x^12 - 1854*x^11 + 3542*x^10 - 3928*x^9 + 7725*x^8 - 8380*x^7 + 17188*x^6 - 17644*x^5 + 8749*x^4 + 7050*x^3 - 20666*x^2 + 3320*x + 7907)
 
gp: K = bnfinit(x^20 + 8*x^18 - 12*x^17 + 60*x^16 - 150*x^15 + 288*x^14 - 666*x^13 + 1043*x^12 - 1854*x^11 + 3542*x^10 - 3928*x^9 + 7725*x^8 - 8380*x^7 + 17188*x^6 - 17644*x^5 + 8749*x^4 + 7050*x^3 - 20666*x^2 + 3320*x + 7907, 1)
 

Normalized defining polynomial

\( x^{20} + 8 x^{18} - 12 x^{17} + 60 x^{16} - 150 x^{15} + 288 x^{14} - 666 x^{13} + 1043 x^{12} - 1854 x^{11} + 3542 x^{10} - 3928 x^{9} + 7725 x^{8} - 8380 x^{7} + 17188 x^{6} - 17644 x^{5} + 8749 x^{4} + 7050 x^{3} - 20666 x^{2} + 3320 x + 7907 \)

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:  \(3522846021083000461302300704768=2^{15}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.68$
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:  $2, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{333964497798916815117661715956355671721836} a^{19} + \frac{10082786121416620058823807374322129753537}{333964497798916815117661715956355671721836} a^{18} - \frac{14567553292976020156297340107085980778895}{333964497798916815117661715956355671721836} a^{17} - \frac{10415350659643230970362338272377883216235}{83491124449729203779415428989088917930459} a^{16} - \frac{18967550120125846808962557263397389439279}{166982248899458407558830857978177835860918} a^{15} - \frac{2153045010309360457740723507747296664677}{19644970458759812653980100938609157160108} a^{14} + \frac{23264564957882134839036880479508781610635}{166982248899458407558830857978177835860918} a^{13} + \frac{9398640466508927969353985825255800516592}{83491124449729203779415428989088917930459} a^{12} + \frac{243902458434633932568456776547773810998}{83491124449729203779415428989088917930459} a^{11} - \frac{2052572905907475191421919650283954479609}{9822485229379906326990050469304578580054} a^{10} - \frac{19348196228721670800386239237993722210405}{333964497798916815117661715956355671721836} a^{9} + \frac{42602840076393173655893597543634270995}{334633765329575967051765246449254180082} a^{8} + \frac{82468884150016373964788197997849000992757}{333964497798916815117661715956355671721836} a^{7} + \frac{15549147170858708560190131181186335806631}{166982248899458407558830857978177835860918} a^{6} + \frac{74416858359843472451994096067575772051297}{166982248899458407558830857978177835860918} a^{5} - \frac{1130154758672315988301440869067960696009}{3630048889118661033887627347351692083933} a^{4} - \frac{16412467814911567794601036337393557746497}{333964497798916815117661715956355671721836} a^{3} - \frac{19369249002115669518768305981356823058080}{83491124449729203779415428989088917930459} a^{2} - \frac{57254912539198837124882496411207453525727}{333964497798916815117661715956355671721836} a - \frac{144161949570901377582292964682520832308927}{333964497798916815117661715956355671721836}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 795087.603907 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:D_4$ (as 20T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 13 conjugacy class representatives for $C_5:D_4$
Character table for $C_5:D_4$

Intermediate fields

\(\Q(\sqrt{401}) \), 4.0.1286408.1, 5.5.160801.1 x5, 10.10.10368641602001.1

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 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
401Data not computed