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Properties

Label 20.0.13610161416...1264.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 7^{10}\cdot 11^{16}$
Root discriminant $36.03$
Ramified primes $2, 7, 11$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, 0, -11520, 0, 59840, 0, -96768, 0, 84004, 0, -42123, 0, 13339, 0, -2709, 0, 352, 0, -27, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 27*x^18 + 352*x^16 - 2709*x^14 + 13339*x^12 - 42123*x^10 + 84004*x^8 - 96768*x^6 + 59840*x^4 - 11520*x^2 + 1024)
 
gp: K = bnfinit(x^20 - 27*x^18 + 352*x^16 - 2709*x^14 + 13339*x^12 - 42123*x^10 + 84004*x^8 - 96768*x^6 + 59840*x^4 - 11520*x^2 + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - 27 x^{18} + 352 x^{16} - 2709 x^{14} + 13339 x^{12} - 42123 x^{10} + 84004 x^{8} - 96768 x^{6} + 59840 x^{4} - 11520 x^{2} + 1024 \)

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:  \(13610161416118240236476132491264=2^{20}\cdot 7^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.03$
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:  $2, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(308=2^{2}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{308}(1,·)$, $\chi_{308}(69,·)$, $\chi_{308}(71,·)$, $\chi_{308}(265,·)$, $\chi_{308}(267,·)$, $\chi_{308}(141,·)$, $\chi_{308}(15,·)$, $\chi_{308}(279,·)$, $\chi_{308}(27,·)$, $\chi_{308}(223,·)$, $\chi_{308}(97,·)$, $\chi_{308}(155,·)$, $\chi_{308}(295,·)$, $\chi_{308}(169,·)$, $\chi_{308}(225,·)$, $\chi_{308}(111,·)$, $\chi_{308}(113,·)$, $\chi_{308}(181,·)$, $\chi_{308}(251,·)$, $\chi_{308}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} + \frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{13} + \frac{1}{8} a^{11} - \frac{1}{2} a^{9} + \frac{3}{8} a^{7} - \frac{1}{8} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{14} + \frac{1}{16} a^{12} - \frac{1}{4} a^{10} - \frac{5}{16} a^{8} - \frac{1}{16} a^{6} - \frac{7}{16} a^{4}$, $\frac{1}{32} a^{15} + \frac{1}{32} a^{13} - \frac{1}{8} a^{11} - \frac{5}{32} a^{9} + \frac{15}{32} a^{7} - \frac{7}{32} a^{5}$, $\frac{1}{1472} a^{16} - \frac{3}{1472} a^{14} + \frac{3}{184} a^{12} + \frac{107}{1472} a^{10} - \frac{541}{1472} a^{8} - \frac{67}{1472} a^{6} + \frac{1}{16} a^{4} + \frac{19}{46} a^{2} - \frac{10}{23}$, $\frac{1}{2944} a^{17} - \frac{3}{2944} a^{15} + \frac{3}{368} a^{13} + \frac{107}{2944} a^{11} - \frac{541}{2944} a^{9} - \frac{67}{2944} a^{7} - \frac{15}{32} a^{5} + \frac{19}{92} a^{3} - \frac{5}{23} a$, $\frac{1}{1583185894022912} a^{18} - \frac{115100223983}{1583185894022912} a^{16} + \frac{5805435085519}{395796473505728} a^{14} - \frac{53492173114581}{1583185894022912} a^{12} + \frac{388525610082623}{1583185894022912} a^{10} - \frac{34741291812711}{1583185894022912} a^{8} - \frac{2996585661244}{6184319898527} a^{6} + \frac{41649316366383}{98949118376432} a^{4} + \frac{1658356294849}{12368639797054} a^{2} + \frac{1701718003759}{6184319898527}$, $\frac{1}{3166371788045824} a^{19} - \frac{115100223983}{3166371788045824} a^{17} + \frac{5805435085519}{791592947011456} a^{15} - \frac{53492173114581}{3166371788045824} a^{13} + \frac{388525610082623}{3166371788045824} a^{11} - \frac{34741291812711}{3166371788045824} a^{9} - \frac{1498292830622}{6184319898527} a^{7} - \frac{57299802010049}{197898236752864} a^{5} + \frac{1658356294849}{24737279594108} a^{3} - \frac{2241300947384}{6184319898527} a$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( -\frac{31296783}{44080853504} a^{19} + \frac{844854913}{44080853504} a^{17} - \frac{2752275337}{11020213376} a^{15} + \frac{84631021531}{44080853504} a^{13} - \frac{415661770801}{44080853504} a^{11} + \frac{56732643007}{1916558848} a^{9} - \frac{80157857823}{1377526672} a^{7} + \frac{177545250757}{2755053344} a^{5} - \frac{3055337375}{86095417} a^{3} + \frac{534189213}{172190834} a \) (order $4$)
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:  \( 4006867.75615 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), \(\Q(i, \sqrt{7})\), \(\Q(\zeta_{11})^+\), 10.0.219503494144.1, 10.10.3689195226078208.1, 10.0.3602729712967.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
7Data not computed
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$