Properties

Label 32.0.445...336.1
Degree $32$
Signature $[0, 16]$
Discriminant $4.458\times 10^{49}$
Root discriminant $35.61$
Ramified primes $2, 3, 61$
Class number $20$ (GRH)
Class group $[2, 10]$ (GRH)
Galois group $C_2^2\times \SL(2,3)$ (as 32T405)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 2*x^30 + 9*x^28 + 4*x^26 + 26*x^24 - 45*x^22 + 10*x^20 - 403*x^18 - 171*x^16 - 1612*x^14 + 160*x^12 - 2880*x^10 + 6656*x^8 + 4096*x^6 + 36864*x^4 + 32768*x^2 + 65536)
 
gp: K = bnfinit(x^32 + 2*x^30 + 9*x^28 + 4*x^26 + 26*x^24 - 45*x^22 + 10*x^20 - 403*x^18 - 171*x^16 - 1612*x^14 + 160*x^12 - 2880*x^10 + 6656*x^8 + 4096*x^6 + 36864*x^4 + 32768*x^2 + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65536, 0, 32768, 0, 36864, 0, 4096, 0, 6656, 0, -2880, 0, 160, 0, -1612, 0, -171, 0, -403, 0, 10, 0, -45, 0, 26, 0, 4, 0, 9, 0, 2, 0, 1]);
 

\( x^{32} + 2 x^{30} + 9 x^{28} + 4 x^{26} + 26 x^{24} - 45 x^{22} + 10 x^{20} - 403 x^{18} - 171 x^{16} - 1612 x^{14} + 160 x^{12} - 2880 x^{10} + 6656 x^{8} + 4096 x^{6} + 36864 x^{4} + 32768 x^{2} + 65536 \)

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:  \(44580746322432044939368312527587514829553937678336\)\(\medspace = 2^{32}\cdot 3^{24}\cdot 61^{16}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $35.61$
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:  $2, 3, 61$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{6} + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{7} + \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{6} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{7} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{14} + \frac{1}{9} a^{12} + \frac{1}{9} a^{10} + \frac{1}{9} a^{8} + \frac{1}{9} a^{6} - \frac{2}{9} a^{4} - \frac{2}{9} a^{2} - \frac{2}{9}$, $\frac{1}{18} a^{17} - \frac{1}{9} a^{15} + \frac{1}{18} a^{13} - \frac{1}{9} a^{11} - \frac{1}{9} a^{9} - \frac{5}{18} a^{7} - \frac{4}{9} a^{5} - \frac{5}{18} a^{3} - \frac{5}{18} a$, $\frac{1}{36} a^{18} - \frac{1}{18} a^{16} + \frac{1}{36} a^{14} + \frac{1}{9} a^{12} - \frac{1}{18} a^{10} - \frac{5}{36} a^{8} - \frac{1}{18} a^{6} + \frac{13}{36} a^{4} + \frac{13}{36} a^{2} - \frac{1}{3}$, $\frac{1}{72} a^{19} - \frac{1}{36} a^{17} + \frac{1}{72} a^{15} - \frac{1}{9} a^{13} + \frac{5}{36} a^{11} - \frac{5}{72} a^{9} - \frac{13}{36} a^{7} - \frac{35}{72} a^{5} - \frac{23}{72} a^{3} + \frac{1}{3} a$, $\frac{1}{144} a^{20} - \frac{1}{72} a^{18} + \frac{1}{144} a^{16} + \frac{1}{9} a^{14} + \frac{5}{72} a^{12} - \frac{5}{144} a^{10} - \frac{1}{72} a^{8} + \frac{37}{144} a^{6} + \frac{49}{144} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{288} a^{21} - \frac{1}{144} a^{19} + \frac{1}{288} a^{17} - \frac{1}{9} a^{15} + \frac{5}{144} a^{13} + \frac{43}{288} a^{11} - \frac{1}{144} a^{9} - \frac{59}{288} a^{7} + \frac{1}{288} a^{5} + \frac{1}{3} a^{3} - \frac{1}{6} a$, $\frac{1}{1728} a^{22} - \frac{1}{864} a^{20} + \frac{1}{1728} a^{18} - \frac{1}{18} a^{16} + \frac{23}{288} a^{14} - \frac{71}{576} a^{12} + \frac{7}{96} a^{10} + \frac{29}{192} a^{8} + \frac{89}{192} a^{6} - \frac{17}{54} a^{4} + \frac{41}{108} a^{2} - \frac{13}{27}$, $\frac{1}{3456} a^{23} - \frac{1}{1728} a^{21} + \frac{1}{3456} a^{19} - \frac{1}{36} a^{17} + \frac{23}{576} a^{15} - \frac{71}{1152} a^{13} - \frac{25}{192} a^{11} - \frac{35}{384} a^{9} + \frac{25}{384} a^{7} - \frac{53}{108} a^{5} - \frac{31}{216} a^{3} - \frac{2}{27} a$, $\frac{1}{6912} a^{24} - \frac{1}{3456} a^{22} + \frac{1}{6912} a^{20} - \frac{1}{72} a^{18} + \frac{23}{1152} a^{16} - \frac{71}{2304} a^{14} + \frac{13}{128} a^{12} - \frac{35}{768} a^{10} + \frac{25}{768} a^{8} - \frac{17}{216} a^{6} + \frac{185}{432} a^{4} - \frac{1}{27} a^{2} - \frac{1}{3}$, $\frac{1}{13824} a^{25} - \frac{1}{6912} a^{23} + \frac{1}{13824} a^{21} - \frac{1}{144} a^{19} + \frac{23}{2304} a^{17} - \frac{71}{4608} a^{15} + \frac{13}{256} a^{13} + \frac{221}{1536} a^{11} + \frac{25}{1536} a^{9} - \frac{89}{432} a^{7} + \frac{41}{864} a^{5} + \frac{13}{27} a^{3} - \frac{1}{2} a$, $\frac{1}{27648} a^{26} - \frac{1}{13824} a^{24} + \frac{1}{27648} a^{22} - \frac{1}{288} a^{20} + \frac{23}{4608} a^{18} - \frac{71}{9216} a^{16} - \frac{217}{1536} a^{14} + \frac{221}{3072} a^{12} - \frac{487}{3072} a^{10} - \frac{89}{864} a^{8} - \frac{823}{1728} a^{6} - \frac{5}{54} a^{4} - \frac{1}{12} a^{2}$, $\frac{1}{55296} a^{27} - \frac{1}{27648} a^{25} + \frac{1}{55296} a^{23} - \frac{1}{576} a^{21} + \frac{23}{9216} a^{19} - \frac{71}{18432} a^{17} - \frac{217}{3072} a^{15} - \frac{803}{6144} a^{13} + \frac{179}{2048} a^{11} - \frac{89}{1728} a^{9} - \frac{247}{3456} a^{7} - \frac{23}{108} a^{5} - \frac{1}{24} a^{3} - \frac{1}{2} a$, $\frac{1}{110592} a^{28} - \frac{1}{55296} a^{26} + \frac{1}{110592} a^{24} - \frac{1}{3456} a^{22} + \frac{5}{55296} a^{20} - \frac{149}{110592} a^{18} - \frac{217}{6144} a^{16} + \frac{287}{4096} a^{14} - \frac{295}{12288} a^{12} + \frac{355}{3456} a^{10} - \frac{1123}{6912} a^{8} - \frac{439}{1728} a^{6} + \frac{23}{432} a^{4} + \frac{19}{54} a^{2} + \frac{2}{27}$, $\frac{1}{221184} a^{29} - \frac{1}{110592} a^{27} + \frac{1}{221184} a^{25} - \frac{1}{6912} a^{23} + \frac{5}{110592} a^{21} - \frac{149}{221184} a^{19} - \frac{217}{12288} a^{17} + \frac{287}{8192} a^{15} - \frac{295}{24576} a^{13} - \frac{797}{6912} a^{11} - \frac{1123}{13824} a^{9} + \frac{137}{3456} a^{7} + \frac{167}{864} a^{5} + \frac{19}{108} a^{3} + \frac{10}{27} a$, $\frac{1}{269402112} a^{30} + \frac{7}{6414336} a^{28} - \frac{613}{89800704} a^{26} - \frac{193}{4810752} a^{24} - \frac{881}{4988928} a^{22} - \frac{5493}{3325952} a^{20} - \frac{1088293}{134701056} a^{18} - \frac{248083}{29933568} a^{16} + \frac{704779}{9977856} a^{14} - \frac{778427}{33675264} a^{12} + \frac{35941}{5612544} a^{10} - \frac{32171}{1403136} a^{8} - \frac{73}{150336} a^{6} + \frac{533}{14616} a^{4} - \frac{367}{1044} a^{2} + \frac{3971}{16443}$, $\frac{1}{538804224} a^{31} + \frac{7}{12828672} a^{29} - \frac{613}{179601408} a^{27} - \frac{193}{9621504} a^{25} - \frac{881}{9977856} a^{23} - \frac{5493}{6651904} a^{21} - \frac{1088293}{269402112} a^{19} - \frac{248083}{59867136} a^{17} - \frac{2621173}{19955712} a^{15} + \frac{10446661}{67350528} a^{13} + \frac{35941}{11225088} a^{11} + \frac{435541}{2806272} a^{9} - \frac{50185}{300672} a^{7} + \frac{533}{29232} a^{5} + \frac{677}{2088} a^{3} + \frac{3971}{32886} a$

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

Class group and class number

$C_{2}\times C_{10}$, which has order $20$ (assuming GRH)

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:  \( \frac{3713}{44900352} a^{31} + \frac{259}{801792} a^{29} + \frac{16141}{44900352} a^{27} + \frac{491}{356352} a^{25} - \frac{27499}{22450176} a^{23} + \frac{123119}{44900352} a^{21} - \frac{205325}{11225088} a^{19} + \frac{1051}{14966784} a^{17} - \frac{1341583}{14966784} a^{15} + \frac{390101}{22450176} a^{13} - \frac{696347}{2806272} a^{11} + \frac{125497}{350784} a^{9} - \frac{7705}{33408} a^{7} + \frac{92447}{43848} a^{5} + \frac{1279}{1566} a^{3} + \frac{31519}{5481} a \) (order $12$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 333896890255.2903 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 333896890255.2903 \cdot 20}{12\sqrt{44580746322432044939368312527587514829553937678336}}\approx 0.491773758057361$ (assuming GRH)

Galois group

$C_2^2\times \SL(2,3)$ (as 32T405):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 96
The 28 conjugacy class representatives for $C_2^2\times \SL(2,3)$
Character table for $C_2^2\times \SL(2,3)$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), 4.4.33489.1, \(\Q(\zeta_{12})\), 8.0.2583966230784.1, 8.0.287107358976.16, 8.8.10093618089.1, 8.8.287107358976.1, 8.0.10093618089.1, 8.0.1121513121.1, 8.8.2583966230784.1, Deg 16, 16.0.6676881481832071949254656.1, Deg 16, Deg 16, 16.0.6676881481832071949254656.3, 16.16.6676881481832071949254656.1, 16.0.101881126126588011921.2

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.6.4.1$x^{6} + 305 x^{3} + 29768$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
61.6.4.1$x^{6} + 305 x^{3} + 29768$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
61.6.4.1$x^{6} + 305 x^{3} + 29768$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
61.6.4.1$x^{6} + 305 x^{3} + 29768$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$