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Properties

Label 32.0.19247509741...0528.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{191}\cdot 23^{16}$
Root discriminant $300.36$
Ramified primes $2, 23$
Class number Not computed
Class group Not computed
Galois group $C_{32}$ (as 32T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12265220831361997297922, 0, 68258620278884158875392, 0, 63065029605490798960960, 0, 23032445595048813533568, 0, 4416936383522559738192, 0, 512108566205514172544, 0, 38964782211289121824, 0, 2047838529976972480, 0, 76793944874136468, 0, 2094975478237312, 0, 41947449507040, 0, 615829298240, 0, 6548279400, 0, 49057344, 0, 245456, 0, 736, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 736*x^30 + 245456*x^28 + 49057344*x^26 + 6548279400*x^24 + 615829298240*x^22 + 41947449507040*x^20 + 2094975478237312*x^18 + 76793944874136468*x^16 + 2047838529976972480*x^14 + 38964782211289121824*x^12 + 512108566205514172544*x^10 + 4416936383522559738192*x^8 + 23032445595048813533568*x^6 + 63065029605490798960960*x^4 + 68258620278884158875392*x^2 + 12265220831361997297922)
 
gp: K = bnfinit(x^32 + 736*x^30 + 245456*x^28 + 49057344*x^26 + 6548279400*x^24 + 615829298240*x^22 + 41947449507040*x^20 + 2094975478237312*x^18 + 76793944874136468*x^16 + 2047838529976972480*x^14 + 38964782211289121824*x^12 + 512108566205514172544*x^10 + 4416936383522559738192*x^8 + 23032445595048813533568*x^6 + 63065029605490798960960*x^4 + 68258620278884158875392*x^2 + 12265220831361997297922, 1)
 

Normalized defining polynomial

\( x^{32} + 736 x^{30} + 245456 x^{28} + 49057344 x^{26} + 6548279400 x^{24} + 615829298240 x^{22} + 41947449507040 x^{20} + 2094975478237312 x^{18} + 76793944874136468 x^{16} + 2047838529976972480 x^{14} + 38964782211289121824 x^{12} + 512108566205514172544 x^{10} + 4416936383522559738192 x^{8} + 23032445595048813533568 x^{6} + 63065029605490798960960 x^{4} + 68258620278884158875392 x^{2} + 12265220831361997297922 \)

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:  \(19247509741360815152884297845798278938249171496527684529937635790938725865750528=2^{191}\cdot 23^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $300.36$
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:  $2, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2944=2^{7}\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{2944}(1,·)$, $\chi_{2944}(2437,·)$, $\chi_{2944}(1289,·)$, $\chi_{2944}(781,·)$, $\chi_{2944}(2577,·)$, $\chi_{2944}(2069,·)$, $\chi_{2944}(921,·)$, $\chi_{2944}(413,·)$, $\chi_{2944}(2209,·)$, $\chi_{2944}(1701,·)$, $\chi_{2944}(553,·)$, $\chi_{2944}(45,·)$, $\chi_{2944}(1841,·)$, $\chi_{2944}(1333,·)$, $\chi_{2944}(185,·)$, $\chi_{2944}(2621,·)$, $\chi_{2944}(1473,·)$, $\chi_{2944}(965,·)$, $\chi_{2944}(2761,·)$, $\chi_{2944}(2253,·)$, $\chi_{2944}(1105,·)$, $\chi_{2944}(597,·)$, $\chi_{2944}(2393,·)$, $\chi_{2944}(1885,·)$, $\chi_{2944}(737,·)$, $\chi_{2944}(229,·)$, $\chi_{2944}(2025,·)$, $\chi_{2944}(1517,·)$, $\chi_{2944}(369,·)$, $\chi_{2944}(2805,·)$, $\chi_{2944}(1657,·)$, $\chi_{2944}(1149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{23} a^{2}$, $\frac{1}{23} a^{3}$, $\frac{1}{529} a^{4}$, $\frac{1}{529} a^{5}$, $\frac{1}{12167} a^{6}$, $\frac{1}{12167} a^{7}$, $\frac{1}{279841} a^{8}$, $\frac{1}{279841} a^{9}$, $\frac{1}{6436343} a^{10}$, $\frac{1}{6436343} a^{11}$, $\frac{1}{148035889} a^{12}$, $\frac{1}{148035889} a^{13}$, $\frac{1}{3404825447} a^{14}$, $\frac{1}{3404825447} a^{15}$, $\frac{1}{78310985281} a^{16}$, $\frac{1}{78310985281} a^{17}$, $\frac{1}{1801152661463} a^{18}$, $\frac{1}{1801152661463} a^{19}$, $\frac{1}{41426511213649} a^{20}$, $\frac{1}{41426511213649} a^{21}$, $\frac{1}{952809757913927} a^{22}$, $\frac{1}{952809757913927} a^{23}$, $\frac{1}{21914624432020321} a^{24}$, $\frac{1}{21914624432020321} a^{25}$, $\frac{1}{504036361936467383} a^{26}$, $\frac{1}{504036361936467383} a^{27}$, $\frac{1}{11592836324538749809} a^{28}$, $\frac{1}{11592836324538749809} a^{29}$, $\frac{1}{266635235464391245607} a^{30}$, $\frac{1}{266635235464391245607} a^{31}$

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:  \( -1 \) (order $2$)
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

$C_{32}$ (as 32T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 32
The 32 conjugacy class representatives for $C_{32}$
Character table for $C_{32}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\), \(\Q(\zeta_{64})^+\)

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 $32$ $32$ $16^{2}$ $32$ $32$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{4}$ $32$ R $32$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ $32$ $32$

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
23Data not computed