Properties

Label 32.0.365...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $3.654\times 10^{49}$
Root discriminant $35.39$
Ramified primes $2, 5, 7$
Class number not computed
Class group not computed
Galois group $C_2^3\times C_4$ (as 32T34)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^28 - 15*x^24 + 31*x^20 + 209*x^16 + 496*x^12 - 3840*x^8 - 4096*x^4 + 65536)
 
gp: K = bnfinit(x^32 - x^28 - 15*x^24 + 31*x^20 + 209*x^16 + 496*x^12 - 3840*x^8 - 4096*x^4 + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65536, 0, 0, 0, -4096, 0, 0, 0, -3840, 0, 0, 0, 496, 0, 0, 0, 209, 0, 0, 0, 31, 0, 0, 0, -15, 0, 0, 0, -1, 0, 0, 0, 1]);
 

\( x^{32} - x^{28} - 15 x^{24} + 31 x^{20} + 209 x^{16} + 496 x^{12} - 3840 x^{8} - 4096 x^{4} + 65536 \)

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

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(36539993586348786372837376000000000000000000000000\)\(\medspace = 2^{64}\cdot 5^{24}\cdot 7^{16}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $35.39$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 5, 7$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $32$
This field is Galois and abelian over $\Q$.
Conductor:  \(280=2^{3}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{280}(1,·)$, $\chi_{280}(139,·)$, $\chi_{280}(13,·)$, $\chi_{280}(279,·)$, $\chi_{280}(153,·)$, $\chi_{280}(27,·)$, $\chi_{280}(29,·)$, $\chi_{280}(69,·)$, $\chi_{280}(167,·)$, $\chi_{280}(41,·)$, $\chi_{280}(43,·)$, $\chi_{280}(181,·)$, $\chi_{280}(183,·)$, $\chi_{280}(111,·)$, $\chi_{280}(57,·)$, $\chi_{280}(267,·)$, $\chi_{280}(197,·)$, $\chi_{280}(71,·)$, $\chi_{280}(141,·)$, $\chi_{280}(209,·)$, $\chi_{280}(83,·)$, $\chi_{280}(223,·)$, $\chi_{280}(97,·)$, $\chi_{280}(99,·)$, $\chi_{280}(237,·)$, $\chi_{280}(239,·)$, $\chi_{280}(113,·)$, $\chi_{280}(211,·)$, $\chi_{280}(169,·)$, $\chi_{280}(251,·)$, $\chi_{280}(253,·)$, $\chi_{280}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{12} + \frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{6} a^{17} + \frac{1}{6} a^{13} + \frac{1}{6} a^{9} + \frac{1}{6} a^{5} + \frac{1}{6} a$, $\frac{1}{12} a^{18} - \frac{5}{12} a^{14} + \frac{1}{12} a^{10} - \frac{5}{12} a^{6} + \frac{1}{12} a^{2}$, $\frac{1}{24} a^{19} + \frac{7}{24} a^{15} + \frac{1}{24} a^{11} + \frac{7}{24} a^{7} + \frac{1}{24} a^{3}$, $\frac{1}{10032} a^{20} - \frac{1}{48} a^{16} + \frac{17}{48} a^{12} - \frac{1}{48} a^{8} + \frac{17}{48} a^{4} + \frac{80}{209}$, $\frac{1}{20064} a^{21} - \frac{1}{96} a^{17} + \frac{17}{96} a^{13} - \frac{1}{96} a^{9} + \frac{17}{96} a^{5} + \frac{40}{209} a$, $\frac{1}{40128} a^{22} - \frac{1}{192} a^{18} - \frac{79}{192} a^{14} + \frac{95}{192} a^{10} + \frac{17}{192} a^{6} - \frac{169}{418} a^{2}$, $\frac{1}{80256} a^{23} - \frac{1}{384} a^{19} + \frac{113}{384} a^{15} - \frac{97}{384} a^{11} - \frac{175}{384} a^{7} - \frac{169}{836} a^{3}$, $\frac{1}{160512} a^{24} - \frac{1}{160512} a^{20} + \frac{11}{256} a^{16} - \frac{91}{256} a^{12} - \frac{85}{256} a^{8} - \frac{3313}{10032} a^{4} - \frac{224}{627}$, $\frac{1}{321024} a^{25} - \frac{1}{321024} a^{21} + \frac{11}{512} a^{17} - \frac{91}{512} a^{13} - \frac{85}{512} a^{9} - \frac{3313}{20064} a^{5} - \frac{112}{627} a$, $\frac{1}{642048} a^{26} - \frac{1}{642048} a^{22} + \frac{11}{1024} a^{18} - \frac{91}{1024} a^{14} - \frac{85}{1024} a^{10} + \frac{16751}{40128} a^{6} - \frac{56}{627} a^{2}$, $\frac{1}{1284096} a^{27} - \frac{1}{1284096} a^{23} + \frac{11}{2048} a^{19} + \frac{933}{2048} a^{15} + \frac{939}{2048} a^{11} + \frac{16751}{80256} a^{7} + \frac{571}{1254} a^{3}$, $\frac{1}{2568192} a^{28} - \frac{1}{2568192} a^{24} - \frac{5}{856064} a^{20} - \frac{529}{12288} a^{16} + \frac{4097}{12288} a^{12} + \frac{17845}{53504} a^{8} + \frac{3329}{10032} a^{4} + \frac{208}{627}$, $\frac{1}{5136384} a^{29} - \frac{1}{5136384} a^{25} - \frac{5}{1712128} a^{21} - \frac{529}{24576} a^{17} + \frac{4097}{24576} a^{13} + \frac{17845}{107008} a^{9} + \frac{3329}{20064} a^{5} + \frac{104}{627} a$, $\frac{1}{10272768} a^{30} - \frac{1}{10272768} a^{26} - \frac{5}{3424256} a^{22} - \frac{529}{49152} a^{18} - \frac{20479}{49152} a^{14} + \frac{17845}{214016} a^{10} - \frac{16735}{40128} a^{6} + \frac{52}{627} a^{2}$, $\frac{1}{20545536} a^{31} - \frac{1}{20545536} a^{27} - \frac{5}{6848512} a^{23} - \frac{529}{98304} a^{19} - \frac{20479}{98304} a^{15} - \frac{196171}{428032} a^{11} - \frac{16735}{80256} a^{7} - \frac{575}{1254} a^{3}$

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

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{17}{321024} a^{29} + \frac{8641}{321024} a^{9} \) (order $40$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{70}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{35}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{7}) \), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{70})\), \(\Q(i, \sqrt{35})\), \(\Q(\sqrt{2}, \sqrt{-35})\), \(\Q(\sqrt{2}, \sqrt{35})\), \(\Q(\sqrt{-2}, \sqrt{35})\), \(\Q(\sqrt{-2}, \sqrt{-35})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{14})\), \(\Q(i, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{7})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{7})\), \(\Q(\sqrt{-2}, \sqrt{-7})\), \(\Q(\sqrt{5}, \sqrt{-14})\), \(\Q(\sqrt{-5}, \sqrt{14})\), \(\Q(\sqrt{-7}, \sqrt{10})\), \(\Q(\sqrt{7}, \sqrt{-10})\), \(\Q(\sqrt{5}, \sqrt{14})\), \(\Q(\sqrt{-5}, \sqrt{-14})\), \(\Q(\sqrt{7}, \sqrt{10})\), \(\Q(\sqrt{-7}, \sqrt{-10})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-5}, \sqrt{7})\), \(\Q(\sqrt{10}, \sqrt{-14})\), \(\Q(\sqrt{-10}, \sqrt{14})\), \(\Q(\sqrt{5}, \sqrt{7})\), \(\Q(\sqrt{-5}, \sqrt{-7})\), \(\Q(\sqrt{10}, \sqrt{14})\), \(\Q(\sqrt{-10}, \sqrt{-14})\), 4.0.392000.2, 4.4.392000.1, 4.0.98000.1, 4.4.6125.1, \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{5})\), 4.4.8000.1, 4.0.8000.2, 8.0.98344960000.9, 8.0.40960000.1, 8.0.157351936.1, 8.0.98344960000.8, 8.0.98344960000.7, 8.0.384160000.1, 8.0.98344960000.2, 8.0.6146560000.2, 8.0.98344960000.3, 8.8.98344960000.1, 8.0.98344960000.1, 8.0.98344960000.6, 8.0.98344960000.5, 8.0.6146560000.1, 8.0.98344960000.4, 8.0.2458624000000.7, 8.0.9604000000.1, \(\Q(\zeta_{20})\), 8.0.1024000000.2, 8.0.2458624000000.6, 8.8.153664000000.1, \(\Q(\zeta_{40})^+\), 8.0.64000000.2, 8.0.153664000000.3, 8.0.2458624000000.5, 8.0.1024000000.1, 8.0.64000000.1, 8.0.2458624000000.4, 8.0.153664000000.5, 8.0.2458624000000.2, 8.0.153664000000.1, 8.0.153664000000.6, 8.8.2458624000000.2, 8.0.2458624000000.1, 8.8.153664000000.2, 8.0.153664000000.4, 8.0.153664000000.2, 8.0.9604000000.2, 8.0.37515625.1, 8.0.2458624000000.3, 8.8.2458624000000.1, 8.0.9604000000.3, 8.8.9604000000.1, 16.0.9671731157401600000000.1, 16.0.6044831973376000000000000.1, \(\Q(\zeta_{40})\), 16.0.6044831973376000000000000.6, 16.0.6044831973376000000000000.4, 16.0.6044831973376000000000000.8, 16.0.92236816000000000000.1, 16.0.6044831973376000000000000.2, 16.0.23612624896000000000000.2, 16.0.6044831973376000000000000.5, 16.16.6044831973376000000000000.1, 16.0.6044831973376000000000000.3, 16.0.6044831973376000000000000.7, 16.0.23612624896000000000000.1, 16.0.6044831973376000000000000.9

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 R ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

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}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$