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Properties

Label 32.0.31714468814...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{64}\cdot 5^{24}\cdot 19^{16}$
Root discriminant $58.30$
Ramified primes $2, 5, 19$
Class number Not computed
Class group Not computed
Galois group $C_2^3\times C_4$ (as 32T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![152587890625, 0, 0, 0, -7568359375, 0, 0, 0, 131250000, 0, 0, 0, 5599375, 0, 0, 0, -487729, 0, 0, 0, 8959, 0, 0, 0, 336, 0, 0, 0, -31, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 31*x^28 + 336*x^24 + 8959*x^20 - 487729*x^16 + 5599375*x^12 + 131250000*x^8 - 7568359375*x^4 + 152587890625)
 
gp: K = bnfinit(x^32 - 31*x^28 + 336*x^24 + 8959*x^20 - 487729*x^16 + 5599375*x^12 + 131250000*x^8 - 7568359375*x^4 + 152587890625, 1)
 

Normalized defining polynomial

\( x^{32} - 31 x^{28} + 336 x^{24} + 8959 x^{20} - 487729 x^{16} + 5599375 x^{12} + 131250000 x^{8} - 7568359375 x^{4} + 152587890625 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(317144688149745561525050153107456000000000000000000000000=2^{64}\cdot 5^{24}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.30$
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:  $2, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(760=2^{3}\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{760}(1,·)$, $\chi_{760}(647,·)$, $\chi_{760}(267,·)$, $\chi_{760}(531,·)$, $\chi_{760}(533,·)$, $\chi_{760}(151,·)$, $\chi_{760}(153,·)$, $\chi_{760}(417,·)$, $\chi_{760}(419,·)$, $\chi_{760}(37,·)$, $\chi_{760}(39,·)$, $\chi_{760}(683,·)$, $\chi_{760}(303,·)$, $\chi_{760}(569,·)$, $\chi_{760}(571,·)$, $\chi_{760}(189,·)$, $\chi_{760}(191,·)$, $\chi_{760}(457,·)$, $\chi_{760}(77,·)$, $\chi_{760}(721,·)$, $\chi_{760}(723,·)$, $\chi_{760}(341,·)$, $\chi_{760}(343,·)$, $\chi_{760}(607,·)$, $\chi_{760}(609,·)$, $\chi_{760}(227,·)$, $\chi_{760}(229,·)$, $\chi_{760}(493,·)$, $\chi_{760}(113,·)$, $\chi_{760}(759,·)$, $\chi_{760}(379,·)$, $\chi_{760}(381,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{9} a^{16} - \frac{2}{9} a^{12} + \frac{4}{9} a^{8} + \frac{1}{9} a^{4} - \frac{2}{9}$, $\frac{1}{45} a^{17} - \frac{11}{45} a^{13} - \frac{14}{45} a^{9} + \frac{19}{45} a^{5} + \frac{16}{45} a$, $\frac{1}{225} a^{18} - \frac{56}{225} a^{14} - \frac{14}{225} a^{10} + \frac{109}{225} a^{6} - \frac{29}{225} a^{2}$, $\frac{1}{1125} a^{19} - \frac{281}{1125} a^{15} + \frac{211}{1125} a^{11} + \frac{334}{1125} a^{7} - \frac{479}{1125} a^{3}$, $\frac{1}{2743475625} a^{20} + \frac{311}{5625} a^{16} + \frac{359}{5625} a^{12} + \frac{2621}{5625} a^{8} - \frac{1876}{5625} a^{4} - \frac{1454228}{4389561}$, $\frac{1}{13717378125} a^{21} + \frac{311}{28125} a^{17} + \frac{359}{28125} a^{13} - \frac{8629}{28125} a^{9} - \frac{13126}{28125} a^{5} + \frac{2935333}{21947805} a$, $\frac{1}{68586890625} a^{22} + \frac{311}{140625} a^{18} + \frac{56609}{140625} a^{14} + \frac{19496}{140625} a^{10} + \frac{14999}{140625} a^{6} + \frac{2935333}{109739025} a^{2}$, $\frac{1}{342934453125} a^{23} + \frac{311}{703125} a^{19} - \frac{84016}{703125} a^{15} + \frac{300746}{703125} a^{11} + \frac{296249}{703125} a^{7} - \frac{216542717}{548695125} a^{3}$, $\frac{1}{1714672265625} a^{24} - \frac{31}{1714672265625} a^{20} - \frac{136516}{3515625} a^{16} + \frac{1693871}{3515625} a^{12} + \frac{1171874}{3515625} a^{8} + \frac{914500834}{2743475625} a^{4} + \frac{487841}{1463187}$, $\frac{1}{8573361328125} a^{25} - \frac{31}{8573361328125} a^{21} - \frac{136516}{17578125} a^{17} - \frac{5337379}{17578125} a^{13} + \frac{4687499}{17578125} a^{9} + \frac{914500834}{13717378125} a^{5} + \frac{1951028}{7315935} a$, $\frac{1}{42866806640625} a^{26} - \frac{31}{42866806640625} a^{22} - \frac{136516}{87890625} a^{18} - \frac{40493629}{87890625} a^{14} + \frac{22265624}{87890625} a^{10} - \frac{12802877291}{68586890625} a^{6} + \frac{16582898}{36579675} a^{2}$, $\frac{1}{214334033203125} a^{27} - \frac{31}{214334033203125} a^{23} - \frac{136516}{439453125} a^{19} + \frac{135287621}{439453125} a^{15} - \frac{153515626}{439453125} a^{11} + \frac{55784013334}{342934453125} a^{7} + \frac{53162573}{182898375} a^{3}$, $\frac{1}{1071670166015625} a^{28} - \frac{31}{1071670166015625} a^{24} + \frac{112}{357223388671875} a^{20} + \frac{65756371}{2197265625} a^{16} + \frac{488281249}{2197265625} a^{12} - \frac{254025517847}{571557421875} a^{8} - \frac{304830289}{2743475625} a^{4} + \frac{975427}{4389561}$, $\frac{1}{5358350830078125} a^{29} - \frac{31}{5358350830078125} a^{25} + \frac{112}{1786116943359375} a^{21} + \frac{65756371}{10986328125} a^{17} - \frac{3906250001}{10986328125} a^{13} - \frac{254025517847}{2857787109375} a^{9} - \frac{304830289}{13717378125} a^{5} + \frac{5364988}{21947805} a$, $\frac{1}{26791754150390625} a^{30} - \frac{31}{26791754150390625} a^{26} + \frac{112}{8930584716796875} a^{22} + \frac{65756371}{54931640625} a^{18} + \frac{7080078124}{54931640625} a^{14} - \frac{3111812627222}{14288935546875} a^{10} + \frac{13412547836}{68586890625} a^{6} + \frac{5364988}{109739025} a^{2}$, $\frac{1}{133958770751953125} a^{31} - \frac{31}{133958770751953125} a^{27} + \frac{112}{44652923583984375} a^{23} + \frac{65756371}{274658203125} a^{19} - \frac{102783203126}{274658203125} a^{15} + \frac{11177122919653}{71444677734375} a^{11} + \frac{13412547836}{342934453125} a^{7} + \frac{5364988}{548695125} a^{3}$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{831041}{133958770751953125} a^{31} + \frac{25762271}{133958770751953125} a^{27} - \frac{93076592}{44652923583984375} a^{23} - \frac{77836}{274658203125} a^{19} + \frac{831041}{274658203125} a^{15} - \frac{7445296319}{214334033203125} a^{11} - \frac{93076592}{114311484375} a^{7} + \frac{25762271}{548695125} a^{3} \) (order $40$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3\times C_4$ (as 32T34):

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 $C_2^3\times C_4$
Character table for $C_2^3\times C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{190}) \), \(\Q(\sqrt{-190}) \), \(\Q(\sqrt{95}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{38}) \), \(\Q(\sqrt{-38}) \), \(\Q(\sqrt{19}) \), \(\Q(\sqrt{-19}) \), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{190})\), \(\Q(i, \sqrt{95})\), \(\Q(\sqrt{2}, \sqrt{95})\), \(\Q(\sqrt{2}, \sqrt{-95})\), \(\Q(\sqrt{-2}, \sqrt{-95})\), \(\Q(\sqrt{-2}, \sqrt{95})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{38})\), \(\Q(i, \sqrt{19})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{19})\), \(\Q(\sqrt{2}, \sqrt{-19})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{-19})\), \(\Q(\sqrt{-2}, \sqrt{19})\), \(\Q(\sqrt{5}, \sqrt{38})\), \(\Q(\sqrt{-5}, \sqrt{-38})\), \(\Q(\sqrt{10}, \sqrt{19})\), \(\Q(\sqrt{-10}, \sqrt{-19})\), \(\Q(\sqrt{5}, \sqrt{-38})\), \(\Q(\sqrt{-5}, \sqrt{38})\), \(\Q(\sqrt{10}, \sqrt{-19})\), \(\Q(\sqrt{-10}, \sqrt{19})\), \(\Q(\sqrt{5}, \sqrt{19})\), \(\Q(\sqrt{-5}, \sqrt{-19})\), \(\Q(\sqrt{10}, \sqrt{38})\), \(\Q(\sqrt{-10}, \sqrt{-38})\), \(\Q(\sqrt{5}, \sqrt{-19})\), \(\Q(\sqrt{-5}, \sqrt{19})\), \(\Q(\sqrt{10}, \sqrt{-38})\), \(\Q(\sqrt{-10}, \sqrt{38})\), \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), 4.0.8000.2, 4.4.8000.1, 4.0.2888000.1, 4.4.2888000.1, 4.0.722000.3, 4.4.45125.1, 8.0.5337948160000.9, 8.0.40960000.1, 8.0.8540717056.1, 8.0.5337948160000.8, 8.0.5337948160000.4, 8.0.20851360000.1, 8.0.5337948160000.3, 8.8.5337948160000.1, 8.0.5337948160000.1, 8.0.333621760000.2, 8.0.5337948160000.2, 8.0.333621760000.1, 8.0.5337948160000.6, 8.0.5337948160000.7, 8.0.5337948160000.5, \(\Q(\zeta_{20})\), 8.0.1024000000.2, 8.0.133448704000000.61, 8.0.521284000000.2, 8.0.64000000.2, \(\Q(\zeta_{40})^+\), 8.0.133448704000000.60, 8.8.8340544000000.2, 8.0.64000000.1, 8.0.1024000000.1, 8.0.8340544000000.4, 8.0.133448704000000.58, 8.0.8340544000000.6, 8.8.133448704000000.3, 8.0.133448704000000.40, 8.8.8340544000000.1, 8.0.8340544000000.5, 8.0.133448704000000.53, 8.0.8340544000000.2, 8.0.133448704000000.42, 8.0.521284000000.3, 8.8.521284000000.1, 8.0.133448704000000.81, 8.8.133448704000000.7, 8.0.2036265625.1, 8.0.521284000000.1, 8.0.8340544000000.1, 8.0.8340544000000.3, 16.0.28493690558847385600000000.1, \(\Q(\zeta_{40})\), 16.0.17808556599279616000000000000.6, 16.0.17808556599279616000000000000.7, 16.0.17808556599279616000000000000.9, 16.0.271737008656000000000000.1, 16.0.17808556599279616000000000000.1, 16.0.17808556599279616000000000000.5, 16.16.17808556599279616000000000000.1, 16.0.69564674215936000000000000.2, 16.0.17808556599279616000000000000.3, 16.0.69564674215936000000000000.1, 16.0.17808556599279616000000000000.8, 16.0.17808556599279616000000000000.4, 16.0.17808556599279616000000000000.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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$5.8.6.1$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}$
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}$
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$