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Properties

Label 32.0.73163329911...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{64}\cdot 5^{24}\cdot 13^{16}$
Root discriminant $48.22$
Ramified primes $2, 5, 13$
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![43046721, 0, 0, 0, -16474671, 0, 0, 0, 5773680, 0, 0, 0, -2006289, 0, 0, 0, 696559, 0, 0, 0, -24769, 0, 0, 0, 880, 0, 0, 0, -31, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 31*x^28 + 880*x^24 - 24769*x^20 + 696559*x^16 - 2006289*x^12 + 5773680*x^8 - 16474671*x^4 + 43046721)
 
gp: K = bnfinit(x^32 - 31*x^28 + 880*x^24 - 24769*x^20 + 696559*x^16 - 2006289*x^12 + 5773680*x^8 - 16474671*x^4 + 43046721, 1)
 

Normalized defining polynomial

\( x^{32} - 31 x^{28} + 880 x^{24} - 24769 x^{20} + 696559 x^{16} - 2006289 x^{12} + 5773680 x^{8} - 16474671 x^{4} + 43046721 \)

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:  \(731633299112184496737658863616000000000000000000000000=2^{64}\cdot 5^{24}\cdot 13^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.22$
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:  $2, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(520=2^{3}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{520}(1,·)$, $\chi_{520}(131,·)$, $\chi_{520}(261,·)$, $\chi_{520}(391,·)$, $\chi_{520}(337,·)$, $\chi_{520}(259,·)$, $\chi_{520}(27,·)$, $\chi_{520}(157,·)$, $\chi_{520}(389,·)$, $\chi_{520}(417,·)$, $\chi_{520}(129,·)$, $\chi_{520}(519,·)$, $\chi_{520}(51,·)$, $\chi_{520}(53,·)$, $\chi_{520}(183,·)$, $\chi_{520}(313,·)$, $\chi_{520}(287,·)$, $\chi_{520}(181,·)$, $\chi_{520}(311,·)$, $\chi_{520}(77,·)$, $\chi_{520}(207,·)$, $\chi_{520}(209,·)$, $\chi_{520}(339,·)$, $\chi_{520}(469,·)$, $\chi_{520}(441,·)$, $\chi_{520}(79,·)$, $\chi_{520}(443,·)$, $\chi_{520}(103,·)$, $\chi_{520}(233,·)$, $\chi_{520}(363,·)$, $\chi_{520}(493,·)$, $\chi_{520}(467,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} + \frac{2}{7} a^{12} - \frac{3}{7} a^{8} + \frac{1}{7} a^{4} + \frac{2}{7}$, $\frac{1}{21} a^{17} + \frac{2}{21} a^{13} + \frac{4}{21} a^{9} + \frac{8}{21} a^{5} - \frac{5}{21} a$, $\frac{1}{63} a^{18} + \frac{23}{63} a^{14} + \frac{25}{63} a^{10} + \frac{8}{63} a^{6} - \frac{5}{63} a^{2}$, $\frac{1}{189} a^{19} + \frac{23}{189} a^{15} - \frac{38}{189} a^{11} + \frac{71}{189} a^{7} - \frac{68}{189} a^{3}$, $\frac{1}{394948953} a^{20} - \frac{19}{567} a^{16} - \frac{59}{567} a^{12} - \frac{34}{567} a^{8} + \frac{163}{567} a^{4} - \frac{2114446}{4875913}$, $\frac{1}{1184846859} a^{21} - \frac{19}{1701} a^{17} + \frac{508}{1701} a^{13} - \frac{601}{1701} a^{9} - \frac{404}{1701} a^{5} - \frac{6990359}{14627739} a$, $\frac{1}{3554540577} a^{22} - \frac{19}{5103} a^{18} + \frac{2209}{5103} a^{14} - \frac{601}{5103} a^{10} - \frac{2105}{5103} a^{6} - \frac{21618098}{43883217} a^{2}$, $\frac{1}{10663621731} a^{23} - \frac{19}{15309} a^{19} + \frac{7312}{15309} a^{15} + \frac{4502}{15309} a^{11} + \frac{2998}{15309} a^{7} - \frac{65501315}{131649651} a^{3}$, $\frac{1}{31990865193} a^{24} - \frac{31}{31990865193} a^{20} + \frac{61}{6561} a^{16} + \frac{2078}{6561} a^{12} + \frac{2812}{6561} a^{8} - \frac{56446048}{394948953} a^{4} - \frac{1392238}{4875913}$, $\frac{1}{95972595579} a^{25} - \frac{31}{95972595579} a^{21} + \frac{61}{19683} a^{17} + \frac{8639}{19683} a^{13} + \frac{2812}{19683} a^{9} + \frac{338502905}{1184846859} a^{5} - \frac{6268151}{14627739} a$, $\frac{1}{287917786737} a^{26} - \frac{31}{287917786737} a^{22} + \frac{61}{59049} a^{18} - \frac{11044}{59049} a^{14} - \frac{16871}{59049} a^{10} + \frac{1523349764}{3554540577} a^{6} - \frac{6268151}{43883217} a^{2}$, $\frac{1}{863753360211} a^{27} - \frac{31}{863753360211} a^{23} + \frac{61}{177147} a^{19} + \frac{48005}{177147} a^{15} - \frac{75920}{177147} a^{11} + \frac{1523349764}{10663621731} a^{7} + \frac{37615066}{131649651} a^{3}$, $\frac{1}{2591260080633} a^{28} - \frac{31}{2591260080633} a^{24} + \frac{880}{2591260080633} a^{20} - \frac{18259}{3720087} a^{16} + \frac{1}{3720087} a^{12} - \frac{24769}{31990865193} a^{8} + \frac{880}{394948953} a^{4} - \frac{31}{4875913}$, $\frac{1}{7773780241899} a^{29} - \frac{31}{7773780241899} a^{25} + \frac{880}{7773780241899} a^{21} - \frac{18259}{11160261} a^{17} + \frac{3720088}{11160261} a^{13} - \frac{31990889962}{95972595579} a^{9} + \frac{394949833}{1184846859} a^{5} - \frac{4875944}{14627739} a$, $\frac{1}{23321340725697} a^{30} - \frac{31}{23321340725697} a^{26} + \frac{880}{23321340725697} a^{22} - \frac{18259}{33480783} a^{18} + \frac{14880349}{33480783} a^{14} + \frac{63981705617}{287917786737} a^{10} + \frac{394949833}{3554540577} a^{6} - \frac{19503683}{43883217} a^{2}$, $\frac{1}{69964022177091} a^{31} - \frac{31}{69964022177091} a^{27} + \frac{880}{69964022177091} a^{23} - \frac{18259}{100442349} a^{19} + \frac{14880349}{100442349} a^{15} + \frac{351899492354}{863753360211} a^{11} + \frac{3949490410}{10663621731} a^{7} - \frac{63386900}{131649651} a^{3}$

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:  \( -\frac{97}{10663621731} a^{27} - \frac{1674257764}{10663621731} a^{7} \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{130}) \), \(\Q(\sqrt{-130}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{-65}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{26}) \), \(\Q(\sqrt{-26}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{130})\), \(\Q(i, \sqrt{65})\), \(\Q(\sqrt{2}, \sqrt{65})\), \(\Q(\sqrt{2}, \sqrt{-65})\), \(\Q(\sqrt{-2}, \sqrt{-65})\), \(\Q(\sqrt{-2}, \sqrt{65})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{26})\), \(\Q(i, \sqrt{13})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{13})\), \(\Q(\sqrt{2}, \sqrt{-13})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{-13})\), \(\Q(\sqrt{-2}, \sqrt{13})\), \(\Q(\sqrt{5}, \sqrt{26})\), \(\Q(\sqrt{-5}, \sqrt{-26})\), \(\Q(\sqrt{10}, \sqrt{13})\), \(\Q(\sqrt{-10}, \sqrt{-13})\), \(\Q(\sqrt{5}, \sqrt{-26})\), \(\Q(\sqrt{-5}, \sqrt{26})\), \(\Q(\sqrt{10}, \sqrt{-13})\), \(\Q(\sqrt{-10}, \sqrt{13})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-5}, \sqrt{-13})\), \(\Q(\sqrt{10}, \sqrt{26})\), \(\Q(\sqrt{-10}, \sqrt{-26})\), \(\Q(\sqrt{5}, \sqrt{-13})\), \(\Q(\sqrt{-5}, \sqrt{13})\), \(\Q(\sqrt{10}, \sqrt{-26})\), \(\Q(\sqrt{-10}, \sqrt{26})\), \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), 4.0.8000.2, 4.4.8000.1, 4.0.1352000.1, 4.4.1352000.1, 4.0.21125.1, 4.4.338000.1, 8.0.1169858560000.10, 8.0.40960000.1, 8.0.1871773696.1, 8.0.1169858560000.9, 8.0.1169858560000.7, 8.0.4569760000.1, 8.0.1169858560000.1, 8.8.73116160000.2, 8.0.1169858560000.4, 8.0.1169858560000.8, 8.0.1169858560000.5, 8.0.1169858560000.6, 8.0.1169858560000.2, 8.0.73116160000.1, 8.0.1169858560000.3, \(\Q(\zeta_{20})\), 8.0.1024000000.2, 8.0.29246464000000.46, 8.0.114244000000.2, 8.0.64000000.2, \(\Q(\zeta_{40})^+\), 8.0.1827904000000.5, 8.8.29246464000000.6, 8.0.64000000.1, 8.0.1024000000.1, 8.0.29246464000000.39, 8.0.1827904000000.1, 8.0.1827904000000.7, 8.8.29246464000000.5, 8.0.1827904000000.3, 8.8.29246464000000.7, 8.0.1827904000000.6, 8.0.29246464000000.38, 8.0.29246464000000.52, 8.0.1827904000000.4, 8.0.446265625.1, 8.8.114244000000.1, 8.0.1827904000000.2, 8.8.1827904000000.1, 8.0.114244000000.3, 8.0.114244000000.1, 8.0.29246464000000.40, 8.0.29246464000000.42, 16.0.1368569050405273600000000.1, \(\Q(\zeta_{40})\), 16.0.855355656503296000000000000.10, 16.0.855355656503296000000000000.11, 16.0.855355656503296000000000000.5, 16.0.13051691536000000000000.1, 16.0.855355656503296000000000000.4, 16.0.3341233033216000000000000.1, 16.16.855355656503296000000000000.2, 16.0.855355656503296000000000000.8, 16.0.855355656503296000000000000.7, 16.0.855355656503296000000000000.9, 16.0.855355656503296000000000000.12, 16.0.3341233033216000000000000.2, 16.0.855355656503296000000000000.6

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$