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Properties

Label 32.32.805...000.2
Degree $32$
Signature $[32, 0]$
Discriminant $8.053\times 10^{81}$
Root discriminant \(362.71\)
Ramified primes $2,3,5$
Class number not computed
Class group not computed
Galois group $C_{32}$ (as 32T33)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 480*x^30 + 104400*x^28 - 13608000*x^26 + 1184625000*x^24 - 72657000000*x^22 + 3227647500000*x^20 - 105129090000000*x^18 + 2513242307812500*x^16 - 43708561875000000*x^14 + 542383517812500000*x^12 - 4649001581250000000*x^10 + 26150633894531250000*x^8 - 88933329843750000000*x^6 + 158809517578125000000*x^4 - 112100835937500000000*x^2 + 25046768244100781250)
 
gp: K = bnfinit(y^32 - 480*y^30 + 104400*y^28 - 13608000*y^26 + 1184625000*y^24 - 72657000000*y^22 + 3227647500000*y^20 - 105129090000000*y^18 + 2513242307812500*y^16 - 43708561875000000*y^14 + 542383517812500000*y^12 - 4649001581250000000*y^10 + 26150633894531250000*y^8 - 88933329843750000000*y^6 + 158809517578125000000*y^4 - 112100835937500000000*y^2 + 25046768244100781250, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 480*x^30 + 104400*x^28 - 13608000*x^26 + 1184625000*x^24 - 72657000000*x^22 + 3227647500000*x^20 - 105129090000000*x^18 + 2513242307812500*x^16 - 43708561875000000*x^14 + 542383517812500000*x^12 - 4649001581250000000*x^10 + 26150633894531250000*x^8 - 88933329843750000000*x^6 + 158809517578125000000*x^4 - 112100835937500000000*x^2 + 25046768244100781250);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 480*x^30 + 104400*x^28 - 13608000*x^26 + 1184625000*x^24 - 72657000000*x^22 + 3227647500000*x^20 - 105129090000000*x^18 + 2513242307812500*x^16 - 43708561875000000*x^14 + 542383517812500000*x^12 - 4649001581250000000*x^10 + 26150633894531250000*x^8 - 88933329843750000000*x^6 + 158809517578125000000*x^4 - 112100835937500000000*x^2 + 25046768244100781250)
 

\( x^{32} - 480 x^{30} + 104400 x^{28} - 13608000 x^{26} + 1184625000 x^{24} - 72657000000 x^{22} + \cdots + 25\!\cdots\!50 \) Copy content Toggle raw display

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

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[32, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(805\!\cdots\!000\) \(\medspace = 2^{191}\cdot 3^{16}\cdot 5^{24}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(362.71\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{191/32}3^{1/2}5^{3/4}\approx 362.71115912243516$
Ramified primes:   \(2\), \(3\), \(5\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{2}) \)
$\card{ \Gal(K/\Q) }$:  $32$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1920=2^{7}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{1920}(1,·)$, $\chi_{1920}(773,·)$, $\chi_{1920}(649,·)$, $\chi_{1920}(1037,·)$, $\chi_{1920}(1681,·)$, $\chi_{1920}(533,·)$, $\chi_{1920}(409,·)$, $\chi_{1920}(797,·)$, $\chi_{1920}(1441,·)$, $\chi_{1920}(293,·)$, $\chi_{1920}(169,·)$, $\chi_{1920}(557,·)$, $\chi_{1920}(1201,·)$, $\chi_{1920}(53,·)$, $\chi_{1920}(1849,·)$, $\chi_{1920}(317,·)$, $\chi_{1920}(961,·)$, $\chi_{1920}(1733,·)$, $\chi_{1920}(1609,·)$, $\chi_{1920}(77,·)$, $\chi_{1920}(721,·)$, $\chi_{1920}(1493,·)$, $\chi_{1920}(1369,·)$, $\chi_{1920}(1757,·)$, $\chi_{1920}(481,·)$, $\chi_{1920}(1253,·)$, $\chi_{1920}(1129,·)$, $\chi_{1920}(1517,·)$, $\chi_{1920}(241,·)$, $\chi_{1920}(1013,·)$, $\chi_{1920}(889,·)$, $\chi_{1920}(1277,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{45}a^{4}$, $\frac{1}{45}a^{5}$, $\frac{1}{135}a^{6}$, $\frac{1}{135}a^{7}$, $\frac{1}{2025}a^{8}$, $\frac{1}{2025}a^{9}$, $\frac{1}{6075}a^{10}$, $\frac{1}{6075}a^{11}$, $\frac{1}{91125}a^{12}$, $\frac{1}{91125}a^{13}$, $\frac{1}{273375}a^{14}$, $\frac{1}{273375}a^{15}$, $\frac{1}{783219375}a^{16}-\frac{16}{52214625}a^{14}-\frac{53}{17404875}a^{12}-\frac{41}{1160325}a^{10}+\frac{74}{386775}a^{8}+\frac{8}{25785}a^{6}-\frac{4}{1719}a^{4}+\frac{22}{573}a^{2}-\frac{87}{191}$, $\frac{1}{675918320625}a^{17}-\frac{1036}{1001360475}a^{15}+\frac{34327}{15020407125}a^{13}-\frac{39769}{1001360475}a^{11}-\frac{16352}{333786825}a^{9}+\frac{21973}{22252455}a^{7}+\frac{25651}{2472495}a^{5}+\frac{5228}{164833}a^{3}-\frac{58533}{164833}a$, $\frac{1}{2027754961875}a^{18}-\frac{2}{15020407125}a^{16}-\frac{67507}{45061221375}a^{14}-\frac{21067}{15020407125}a^{12}-\frac{9448}{1001360475}a^{10}+\frac{9028}{66757365}a^{8}-\frac{1369}{824165}a^{6}+\frac{33544}{7417485}a^{4}-\frac{42136}{494499}a^{2}+\frac{3}{191}$, $\frac{1}{2027754961875}a^{19}+\frac{22351}{45061221375}a^{15}-\frac{63464}{15020407125}a^{13}+\frac{12556}{333786825}a^{11}+\frac{56957}{333786825}a^{9}-\frac{12463}{7417485}a^{7}+\frac{36328}{7417485}a^{5}+\frac{16920}{164833}a^{3}+\frac{9275}{164833}a$, $\frac{1}{30416324428125}a^{20}-\frac{29}{225306106875}a^{16}-\frac{5476}{5006802375}a^{14}+\frac{73051}{15020407125}a^{12}-\frac{24682}{1001360475}a^{10}-\frac{49471}{333786825}a^{8}-\frac{40372}{22252455}a^{6}+\frac{5021}{7417485}a^{4}+\frac{906}{164833}a^{2}-\frac{30}{191}$, $\frac{1}{30416324428125}a^{21}+\frac{15601}{45061221375}a^{15}-\frac{24109}{5006802375}a^{13}-\frac{23092}{1001360475}a^{11}+\frac{11402}{333786825}a^{9}+\frac{19372}{7417485}a^{7}-\frac{6469}{824165}a^{5}+\frac{48562}{494499}a^{3}-\frac{8438}{164833}a$, $\frac{1}{91248973284375}a^{22}+\frac{67}{135183664125}a^{16}+\frac{16562}{45061221375}a^{14}+\frac{45058}{15020407125}a^{12}+\frac{12809}{200272095}a^{10}-\frac{5837}{111262275}a^{8}-\frac{20249}{22252455}a^{6}-\frac{5651}{2472495}a^{4}-\frac{68848}{494499}a^{2}-\frac{1}{191}$, $\frac{1}{91248973284375}a^{23}-\frac{8291}{15020407125}a^{15}-\frac{16202}{3004081425}a^{13}+\frac{11729}{333786825}a^{11}+\frac{4184}{66757365}a^{9}+\frac{36281}{22252455}a^{7}-\frac{1828}{164833}a^{5}-\frac{48332}{494499}a^{3}-\frac{7435}{164833}a$, $\frac{1}{13\!\cdots\!25}a^{24}+\frac{154}{675918320625}a^{16}+\frac{5191}{3004081425}a^{14}+\frac{1828}{1001360475}a^{12}-\frac{439}{13351473}a^{10}+\frac{75116}{333786825}a^{8}+\frac{18931}{22252455}a^{6}-\frac{54373}{7417485}a^{4}+\frac{54608}{494499}a^{2}-\frac{40}{191}$, $\frac{1}{13\!\cdots\!25}a^{25}+\frac{4693}{45061221375}a^{15}+\frac{15718}{15020407125}a^{13}-\frac{488}{66757365}a^{11}-\frac{14668}{111262275}a^{9}-\frac{68251}{22252455}a^{7}-\frac{37159}{7417485}a^{5}-\frac{53066}{494499}a^{3}+\frac{78580}{164833}a$, $\frac{1}{41\!\cdots\!75}a^{26}+\frac{164}{675918320625}a^{16}+\frac{58868}{45061221375}a^{14}+\frac{44522}{15020407125}a^{12}-\frac{8629}{111262275}a^{10}+\frac{3092}{13351473}a^{8}-\frac{6526}{2472495}a^{6}+\frac{35857}{7417485}a^{4}+\frac{60457}{494499}a^{2}+\frac{57}{191}$, $\frac{1}{41\!\cdots\!75}a^{27}-\frac{14201}{15020407125}a^{15}+\frac{19216}{15020407125}a^{13}+\frac{15968}{1001360475}a^{11}-\frac{43133}{333786825}a^{9}-\frac{7196}{4450491}a^{7}-\frac{57127}{7417485}a^{5}-\frac{39224}{494499}a^{3}-\frac{76544}{164833}a$, $\frac{1}{61\!\cdots\!25}a^{28}-\frac{316}{675918320625}a^{16}-\frac{1058}{600816285}a^{14}-\frac{82414}{15020407125}a^{12}-\frac{61097}{1001360475}a^{10}-\frac{38569}{333786825}a^{8}-\frac{35762}{22252455}a^{6}-\frac{60799}{7417485}a^{4}+\frac{24907}{494499}a^{2}-\frac{61}{191}$, $\frac{1}{61\!\cdots\!25}a^{29}+\frac{316}{600816285}a^{15}+\frac{50773}{15020407125}a^{13}+\frac{12808}{200272095}a^{11}+\frac{13771}{66757365}a^{9}-\frac{16}{23301}a^{7}+\frac{25898}{7417485}a^{5}+\frac{36061}{494499}a^{3}+\frac{77058}{164833}a$, $\frac{1}{18\!\cdots\!75}a^{30}+\frac{269}{675918320625}a^{16}-\frac{35527}{45061221375}a^{14}-\frac{6877}{15020407125}a^{12}-\frac{28664}{1001360475}a^{10}+\frac{75488}{333786825}a^{8}+\frac{7672}{2472495}a^{6}+\frac{14486}{1483497}a^{4}-\frac{51529}{494499}a^{2}+\frac{77}{191}$, $\frac{1}{18\!\cdots\!75}a^{31}-\frac{4411}{9012244275}a^{15}-\frac{10192}{15020407125}a^{13}-\frac{44948}{1001360475}a^{11}+\frac{1579}{22252455}a^{9}-\frac{24178}{7417485}a^{7}-\frac{7934}{2472495}a^{5}+\frac{15133}{494499}a^{3}-\frac{12140}{164833}a$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $31$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 480*x^30 + 104400*x^28 - 13608000*x^26 + 1184625000*x^24 - 72657000000*x^22 + 3227647500000*x^20 - 105129090000000*x^18 + 2513242307812500*x^16 - 43708561875000000*x^14 + 542383517812500000*x^12 - 4649001581250000000*x^10 + 26150633894531250000*x^8 - 88933329843750000000*x^6 + 158809517578125000000*x^4 - 112100835937500000000*x^2 + 25046768244100781250)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^32 - 480*x^30 + 104400*x^28 - 13608000*x^26 + 1184625000*x^24 - 72657000000*x^22 + 3227647500000*x^20 - 105129090000000*x^18 + 2513242307812500*x^16 - 43708561875000000*x^14 + 542383517812500000*x^12 - 4649001581250000000*x^10 + 26150633894531250000*x^8 - 88933329843750000000*x^6 + 158809517578125000000*x^4 - 112100835937500000000*x^2 + 25046768244100781250, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^32 - 480*x^30 + 104400*x^28 - 13608000*x^26 + 1184625000*x^24 - 72657000000*x^22 + 3227647500000*x^20 - 105129090000000*x^18 + 2513242307812500*x^16 - 43708561875000000*x^14 + 542383517812500000*x^12 - 4649001581250000000*x^10 + 26150633894531250000*x^8 - 88933329843750000000*x^6 + 158809517578125000000*x^4 - 112100835937500000000*x^2 + 25046768244100781250);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 480*x^30 + 104400*x^28 - 13608000*x^26 + 1184625000*x^24 - 72657000000*x^22 + 3227647500000*x^20 - 105129090000000*x^18 + 2513242307812500*x^16 - 43708561875000000*x^14 + 542383517812500000*x^12 - 4649001581250000000*x^10 + 26150633894531250000*x^8 - 88933329843750000000*x^6 + 158809517578125000000*x^4 - 112100835937500000000*x^2 + 25046768244100781250);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A cyclic group of order 32
The 32 conjugacy class representatives for $C_{32}$
Character table for $C_{32}$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\), 16.16.236118324143482260684800000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

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

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $32$$32$$1$$191$
\(3\) Copy content Toggle raw display Deg $32$$2$$16$$16$
\(5\) Copy content Toggle raw display Deg $32$$4$$8$$24$