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Properties

Label 32.0.98373307738...4336.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{96}\cdot 3^{16}\cdot 19^{16}$
Root discriminant $60.40$
Ramified primes $2, 3, 19$
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, 0, 0, 0, 0, 112890625, 0, 0, 0, 0, 0, 0, 0, -307104, 0, 0, 0, 0, 0, 0, 0, 289, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 289*x^24 - 307104*x^16 + 112890625*x^8 + 152587890625)
 
gp: K = bnfinit(x^32 + 289*x^24 - 307104*x^16 + 112890625*x^8 + 152587890625, 1)
 

Normalized defining polynomial

\( x^{32} + 289 x^{24} - 307104 x^{16} + 112890625 x^{8} + 152587890625 \)

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:  \(983733077380442711194506053245519508393843121594707214336=2^{96}\cdot 3^{16}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.40$
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:  $2, 3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(912=2^{4}\cdot 3\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{912}(1,·)$, $\chi_{912}(647,·)$, $\chi_{912}(265,·)$, $\chi_{912}(911,·)$, $\chi_{912}(533,·)$, $\chi_{912}(151,·)$, $\chi_{912}(797,·)$, $\chi_{912}(799,·)$, $\chi_{912}(419,·)$, $\chi_{912}(37,·)$, $\chi_{912}(683,·)$, $\chi_{912}(685,·)$, $\chi_{912}(305,·)$, $\chi_{912}(569,·)$, $\chi_{912}(571,·)$, $\chi_{912}(191,·)$, $\chi_{912}(835,·)$, $\chi_{912}(455,·)$, $\chi_{912}(457,·)$, $\chi_{912}(77,·)$, $\chi_{912}(721,·)$, $\chi_{912}(341,·)$, $\chi_{912}(343,·)$, $\chi_{912}(607,·)$, $\chi_{912}(227,·)$, $\chi_{912}(229,·)$, $\chi_{912}(875,·)$, $\chi_{912}(493,·)$, $\chi_{912}(113,·)$, $\chi_{912}(115,·)$, $\chi_{912}(761,·)$, $\chi_{912}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{279} a^{16} + \frac{5}{279} a^{8} + \frac{25}{279}$, $\frac{1}{1395} a^{17} + \frac{284}{1395} a^{9} - \frac{254}{1395} a$, $\frac{1}{6975} a^{18} - \frac{1111}{6975} a^{10} - \frac{254}{6975} a^{2}$, $\frac{1}{34875} a^{19} - \frac{8086}{34875} a^{11} - \frac{7229}{34875} a^{3}$, $\frac{1}{174375} a^{20} - \frac{77836}{174375} a^{12} - \frac{42104}{174375} a^{4}$, $\frac{1}{871875} a^{21} - \frac{77836}{871875} a^{13} - \frac{216479}{871875} a^{5}$, $\frac{1}{4359375} a^{22} - \frac{77836}{4359375} a^{14} - \frac{1088354}{4359375} a^{6}$, $\frac{1}{21796875} a^{23} - \frac{77836}{21796875} a^{15} - \frac{1088354}{21796875} a^{7}$, $\frac{1}{33469537500000} a^{24} + \frac{136516}{108984375} a^{16} + \frac{32812499}{108984375} a^{8} - \frac{42380063}{85682016}$, $\frac{1}{167347687500000} a^{25} + \frac{136516}{544921875} a^{17} + \frac{250781249}{544921875} a^{9} - \frac{128062079}{428410080} a$, $\frac{1}{836738437500000} a^{26} + \frac{136516}{2724609375} a^{18} + \frac{1340624999}{2724609375} a^{10} + \frac{728758081}{2142050400} a^{2}$, $\frac{1}{4183692187500000} a^{27} + \frac{136516}{13623046875} a^{19} - \frac{4108593751}{13623046875} a^{11} - \frac{3555342719}{10710252000} a^{3}$, $\frac{1}{20918460937500000} a^{28} + \frac{136516}{68115234375} a^{20} - \frac{17731640626}{68115234375} a^{12} + \frac{7154909281}{53551260000} a^{4}$, $\frac{1}{104592304687500000} a^{29} + \frac{136516}{340576171875} a^{21} - \frac{153962109376}{340576171875} a^{13} - \frac{46396350719}{267756300000} a^{5}$, $\frac{1}{522961523437500000} a^{30} + \frac{136516}{1702880859375} a^{22} + \frac{186614062499}{1702880859375} a^{14} + \frac{221359949281}{1338781500000} a^{6}$, $\frac{1}{2614807617187500000} a^{31} + \frac{136516}{8514404296875} a^{23} + \frac{3592375781249}{8514404296875} a^{15} - \frac{2456203050719}{6693907500000} a^{7}$

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:  \( -\frac{1}{6693907500000} a^{31} + \frac{23790856319}{6693907500000} a^{7} \) (order $48$)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{19}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-114}) \), \(\Q(\sqrt{114}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{38}) \), \(\Q(\sqrt{-38}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-57}) \), \(\Q(\sqrt{57}) \), \(\Q(i, \sqrt{19})\), \(\Q(i, \sqrt{6})\), \(\Q(i, \sqrt{114})\), \(\Q(\sqrt{-6}, \sqrt{19})\), \(\Q(\sqrt{6}, \sqrt{19})\), \(\Q(\sqrt{-6}, \sqrt{-19})\), \(\Q(\sqrt{6}, \sqrt{-19})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{38})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{57})\), \(\Q(\sqrt{2}, \sqrt{19})\), \(\Q(\sqrt{-2}, \sqrt{19})\), \(\Q(\sqrt{-3}, \sqrt{19})\), \(\Q(\sqrt{3}, \sqrt{19})\), \(\Q(\sqrt{2}, \sqrt{-19})\), \(\Q(\sqrt{-2}, \sqrt{-19})\), \(\Q(\sqrt{-3}, \sqrt{-19})\), \(\Q(\sqrt{3}, \sqrt{-19})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{-6}, \sqrt{38})\), \(\Q(\sqrt{-6}, \sqrt{-38})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{6}, \sqrt{38})\), \(\Q(\sqrt{6}, \sqrt{-38})\), \(\Q(\sqrt{2}, \sqrt{-57})\), \(\Q(\sqrt{-2}, \sqrt{57})\), \(\Q(\sqrt{-3}, \sqrt{38})\), \(\Q(\sqrt{3}, \sqrt{-38})\), \(\Q(\sqrt{2}, \sqrt{57})\), \(\Q(\sqrt{-2}, \sqrt{-57})\), \(\Q(\sqrt{3}, \sqrt{38})\), \(\Q(\sqrt{-3}, \sqrt{-38})\), 4.4.6653952.1, 4.0.6653952.2, 4.4.18432.1, 4.0.18432.2, 4.0.739328.2, 4.4.739328.1, 4.0.2048.2, \(\Q(\zeta_{16})^+\), 8.0.691798081536.3, 8.0.8540717056.1, 8.0.2702336256.1, \(\Q(\zeta_{24})\), 8.0.691798081536.2, 8.0.691798081536.8, 8.0.691798081536.9, 8.0.691798081536.7, 8.0.691798081536.4, 8.8.691798081536.1, 8.0.691798081536.6, 8.0.43237380096.5, 8.0.691798081536.1, 8.0.691798081536.5, 8.0.43237380096.3, 8.0.177100308873216.72, 8.0.1358954496.4, 8.0.2186423566336.2, \(\Q(\zeta_{16})\), 8.8.177100308873216.2, 8.0.177100308873216.69, 8.0.2186423566336.1, 8.8.2186423566336.1, 8.0.44275077218304.5, 8.0.44275077218304.41, 8.0.546605891584.2, 8.0.546605891584.1, 8.0.44275077218304.105, 8.0.44275077218304.108, 8.0.339738624.1, 8.0.339738624.2, 8.8.177100308873216.3, 8.0.177100308873216.51, \(\Q(\zeta_{48})^+\), 8.0.1358954496.3, 8.0.177100308873216.53, 8.0.177100308873216.55, 8.0.177100308873216.38, 8.0.177100308873216.37, 8.8.44275077218304.4, 8.0.44275077218304.60, 8.8.44275077218304.3, 8.0.44275077218304.15, 16.0.478584585616890104119296.1, 16.0.31364519402988509863562182656.1, 16.0.4780448011429432992464896.1, 16.0.31364519402988509863562182656.7, \(\Q(\zeta_{48})\), 16.0.31364519402988509863562182656.6, 16.0.31364519402988509863562182656.9, 16.0.31364519402988509863562182656.2, 16.0.31364519402988509863562182656.3, 16.16.31364519402988509863562182656.1, 16.0.31364519402988509863562182656.4, 16.0.1960282462686781866472636416.1, 16.0.1960282462686781866472636416.2, 16.0.31364519402988509863562182656.8, 16.0.31364519402988509863562182656.5

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$