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Properties

Label 32.32.2251280432...4576.1
Degree $32$
Signature $[32, 0]$
Discriminant $2^{64}\cdot 3^{16}\cdot 17^{28}$
Root discriminant $82.65$
Ramified primes $2, 3, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2\times C_8$ (as 32T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -144, 0, 4236, 0, -54264, 0, 372978, 0, -1521784, 0, 3912050, 0, -6560964, 0, 7298927, 0, -5396088, 0, 2624266, 0, -824692, 0, 164371, 0, -20332, 0, 1499, 0, -60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 60*x^30 + 1499*x^28 - 20332*x^26 + 164371*x^24 - 824692*x^22 + 2624266*x^20 - 5396088*x^18 + 7298927*x^16 - 6560964*x^14 + 3912050*x^12 - 1521784*x^10 + 372978*x^8 - 54264*x^6 + 4236*x^4 - 144*x^2 + 1)
 
gp: K = bnfinit(x^32 - 60*x^30 + 1499*x^28 - 20332*x^26 + 164371*x^24 - 824692*x^22 + 2624266*x^20 - 5396088*x^18 + 7298927*x^16 - 6560964*x^14 + 3912050*x^12 - 1521784*x^10 + 372978*x^8 - 54264*x^6 + 4236*x^4 - 144*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{32} - 60 x^{30} + 1499 x^{28} - 20332 x^{26} + 164371 x^{24} - 824692 x^{22} + 2624266 x^{20} - 5396088 x^{18} + 7298927 x^{16} - 6560964 x^{14} + 3912050 x^{12} - 1521784 x^{10} + 372978 x^{8} - 54264 x^{6} + 4236 x^{4} - 144 x^{2} + 1 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[32, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(22512804325029881197071829100162696243017892592340199408664576=2^{64}\cdot 3^{16}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.65$
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:  $2, 3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(408=2^{3}\cdot 3\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{408}(1,·)$, $\chi_{408}(263,·)$, $\chi_{408}(395,·)$, $\chi_{408}(13,·)$, $\chi_{408}(145,·)$, $\chi_{408}(407,·)$, $\chi_{408}(25,·)$, $\chi_{408}(155,·)$, $\chi_{408}(157,·)$, $\chi_{408}(287,·)$, $\chi_{408}(35,·)$, $\chi_{408}(169,·)$, $\chi_{408}(47,·)$, $\chi_{408}(49,·)$, $\chi_{408}(179,·)$, $\chi_{408}(59,·)$, $\chi_{408}(191,·)$, $\chi_{408}(325,·)$, $\chi_{408}(203,·)$, $\chi_{408}(205,·)$, $\chi_{408}(83,·)$, $\chi_{408}(217,·)$, $\chi_{408}(349,·)$, $\chi_{408}(229,·)$, $\chi_{408}(359,·)$, $\chi_{408}(361,·)$, $\chi_{408}(239,·)$, $\chi_{408}(373,·)$, $\chi_{408}(121,·)$, $\chi_{408}(251,·)$, $\chi_{408}(253,·)$, $\chi_{408}(383,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{1376989282023370048884303555857} a^{30} + \frac{234308147599352788948111778700}{1376989282023370048884303555857} a^{28} - \frac{528948077814056637462407446949}{1376989282023370048884303555857} a^{26} + \frac{619080091695451069396916607759}{1376989282023370048884303555857} a^{24} + \frac{162978238519651622995282499058}{1376989282023370048884303555857} a^{22} - \frac{120034759521937455913202374570}{1376989282023370048884303555857} a^{20} - \frac{463107010894690891078547606111}{1376989282023370048884303555857} a^{18} + \frac{511031766377964467000886590469}{1376989282023370048884303555857} a^{16} - \frac{434213372932914990026551014778}{1376989282023370048884303555857} a^{14} + \frac{3840578907015763896534067424}{1376989282023370048884303555857} a^{12} - \frac{488499709502116080154665060205}{1376989282023370048884303555857} a^{10} + \frac{285271297827859953648546334486}{1376989282023370048884303555857} a^{8} + \frac{611021969852724458666044782726}{1376989282023370048884303555857} a^{6} - \frac{214906973432998705173058126004}{1376989282023370048884303555857} a^{4} + \frac{544146818797505086776552574478}{1376989282023370048884303555857} a^{2} - \frac{368178916027983813859465303707}{1376989282023370048884303555857}$, $\frac{1}{1376989282023370048884303555857} a^{31} + \frac{234308147599352788948111778700}{1376989282023370048884303555857} a^{29} - \frac{528948077814056637462407446949}{1376989282023370048884303555857} a^{27} + \frac{619080091695451069396916607759}{1376989282023370048884303555857} a^{25} + \frac{162978238519651622995282499058}{1376989282023370048884303555857} a^{23} - \frac{120034759521937455913202374570}{1376989282023370048884303555857} a^{21} - \frac{463107010894690891078547606111}{1376989282023370048884303555857} a^{19} + \frac{511031766377964467000886590469}{1376989282023370048884303555857} a^{17} - \frac{434213372932914990026551014778}{1376989282023370048884303555857} a^{15} + \frac{3840578907015763896534067424}{1376989282023370048884303555857} a^{13} - \frac{488499709502116080154665060205}{1376989282023370048884303555857} a^{11} + \frac{285271297827859953648546334486}{1376989282023370048884303555857} a^{9} + \frac{611021969852724458666044782726}{1376989282023370048884303555857} a^{7} - \frac{214906973432998705173058126004}{1376989282023370048884303555857} a^{5} + \frac{544146818797505086776552574478}{1376989282023370048884303555857} a^{3} - \frac{368178916027983813859465303707}{1376989282023370048884303555857} a$

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:  $31$
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:  \( 151446630849813740000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_8$ (as 32T37):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2^2\times C_8$
Character table for $C_2^2\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{51}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{102}) \), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{17})\), \(\Q(\sqrt{3}, \sqrt{34})\), \(\Q(\sqrt{2}, \sqrt{17})\), \(\Q(\sqrt{2}, \sqrt{51})\), \(\Q(\sqrt{6}, \sqrt{17})\), \(\Q(\sqrt{6}, \sqrt{34})\), 4.4.4913.1, 4.4.707472.1, 4.4.314432.1, 4.4.2829888.2, 8.8.443364212736.1, 8.8.500516630784.1, 8.8.128132257480704.2, 8.8.98867482624.1, 8.8.128132257480704.3, 8.8.8008266092544.1, 8.8.128132257480704.1, 8.8.1680747204608.1, 8.8.136140523573248.1, \(\Q(\zeta_{17})^+\), 8.8.8508782723328.1, 16.16.16417875407101426168932335616.1, 16.16.4744765992652312162821444993024.2, 16.16.72399383432805056195395584.1, 16.16.2824911165797606216433664.1, 16.16.4744765992652312162821444993024.1, 16.16.4744765992652312162821444993024.3, 16.16.18534242158798094386021269504.1

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{16}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
3Data not computed
17Data not computed