Properties

Label 32.0.805...000.1
Degree $32$
Signature $[0, 16]$
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 + 1226865178750781250)
 
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 + 1226865178750781250, 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 + 1226865178750781250);
 
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 + 1226865178750781250)
 

\( x^{32} + 480 x^{30} + 104400 x^{28} + 13608000 x^{26} + 1184625000 x^{24} + 72657000000 x^{22} + 3227647500000 x^{20} + 105129090000000 x^{18} + \cdots + 12\!\cdots\!50 \) 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:  $[0, 16]$
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$$\chi_{1920}(1,·)$, $\chi_{1920}(1283,·)$, $\chi_{1920}(649,·)$, $\chi_{1920}(1547,·)$, $\chi_{1920}(1681,·)$, $\chi_{1920}(1043,·)$, $\chi_{1920}(409,·)$, $\chi_{1920}(1307,·)$, $\chi_{1920}(1441,·)$, $\chi_{1920}(803,·)$, $\chi_{1920}(169,·)$, $\chi_{1920}(1067,·)$, $\chi_{1920}(1201,·)$, $\chi_{1920}(563,·)$, $\chi_{1920}(1849,·)$, $\chi_{1920}(827,·)$, $\chi_{1920}(961,·)$, $\chi_{1920}(323,·)$, $\chi_{1920}(1609,·)$, $\chi_{1920}(587,·)$, $\chi_{1920}(721,·)$, $\chi_{1920}(83,·)$, $\chi_{1920}(1369,·)$, $\chi_{1920}(347,·)$, $\chi_{1920}(481,·)$, $\chi_{1920}(1763,·)$, $\chi_{1920}(1129,·)$, $\chi_{1920}(107,·)$, $\chi_{1920}(241,·)$, $\chi_{1920}(1523,·)$, $\chi_{1920}(889,·)$, $\chi_{1920}(1787,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{45}a^{4}$, $\frac{1}{45}a^{5}$, $\frac{1}{135}a^{6}$, $\frac{1}{135}a^{7}$, $\frac{1}{2025}a^{8}$, $\frac{1}{2025}a^{9}$, $\frac{1}{6075}a^{10}$, $\frac{1}{6075}a^{11}$, $\frac{1}{91125}a^{12}$, $\frac{1}{91125}a^{13}$, $\frac{1}{273375}a^{14}$, $\frac{1}{273375}a^{15}$, $\frac{1}{3538839375}a^{16}+\frac{16}{235922625}a^{14}-\frac{343}{78640875}a^{12}+\frac{34}{5242725}a^{10}+\frac{103}{1747575}a^{8}+\frac{403}{116505}a^{6}-\frac{287}{38835}a^{4}+\frac{233}{2589}a^{2}+\frac{387}{863}$, $\frac{1}{675918320625}a^{17}-\frac{46586}{45061221375}a^{15}-\frac{33137}{15020407125}a^{13}-\frac{11783}{333786825}a^{11}+\frac{21034}{111262275}a^{9}+\frac{78073}{22252455}a^{7}-\frac{63286}{7417485}a^{5}+\frac{22228}{164833}a^{3}+\frac{79783}{164833}a$, $\frac{1}{2027754961875}a^{18}+\frac{2}{15020407125}a^{16}+\frac{68857}{45061221375}a^{14}+\frac{6233}{15020407125}a^{12}+\frac{73798}{1001360475}a^{10}-\frac{3859}{333786825}a^{8}+\frac{53797}{22252455}a^{6}+\frac{5468}{824165}a^{4}-\frac{7149}{164833}a^{2}+\frac{79}{863}$, $\frac{1}{2027754961875}a^{19}-\frac{24061}{45061221375}a^{15}+\frac{21569}{15020407125}a^{13}-\frac{41452}{1001360475}a^{11}-\frac{26239}{111262275}a^{9}-\frac{49787}{22252455}a^{7}-\frac{24203}{7417485}a^{5}+\frac{75814}{494499}a^{3}-\frac{77562}{164833}a$, $\frac{1}{30416324428125}a^{20}+\frac{1}{135183664125}a^{16}-\frac{39196}{45061221375}a^{14}-\frac{10888}{3004081425}a^{12}-\frac{54631}{1001360475}a^{10}-\frac{43484}{333786825}a^{8}+\frac{1577}{22252455}a^{6}-\frac{4865}{494499}a^{4}-\frac{15141}{164833}a^{2}-\frac{429}{863}$, $\frac{1}{30416324428125}a^{21}+\frac{28901}{45061221375}a^{15}-\frac{53588}{15020407125}a^{13}-\frac{42719}{1001360475}a^{11}-\frac{9776}{111262275}a^{9}-\frac{59122}{22252455}a^{7}+\frac{78622}{7417485}a^{5}-\frac{49177}{494499}a^{3}+\frac{13645}{164833}a$, $\frac{1}{91248973284375}a^{22}-\frac{82}{675918320625}a^{16}-\frac{59318}{45061221375}a^{14}-\frac{70154}{15020407125}a^{12}-\frac{296}{1001360475}a^{10}-\frac{4687}{66757365}a^{8}-\frac{778}{4450491}a^{6}+\frac{42334}{7417485}a^{4}-\frac{16398}{164833}a^{2}-\frac{402}{863}$, $\frac{1}{91248973284375}a^{23}+\frac{76622}{45061221375}a^{15}+\frac{14773}{15020407125}a^{13}+\frac{13616}{200272095}a^{11}+\frac{13702}{111262275}a^{9}-\frac{30391}{22252455}a^{7}-\frac{7459}{1483497}a^{5}-\frac{6865}{164833}a^{3}+\frac{36937}{164833}a$, $\frac{1}{13\!\cdots\!25}a^{24}+\frac{31}{675918320625}a^{16}+\frac{63196}{45061221375}a^{14}-\frac{34487}{15020407125}a^{12}-\frac{24478}{1001360475}a^{10}-\frac{1456}{66757365}a^{8}-\frac{49861}{22252455}a^{6}+\frac{38233}{7417485}a^{4}-\frac{3385}{494499}a^{2}+\frac{153}{863}$, $\frac{1}{13\!\cdots\!25}a^{25}+\frac{1591}{3004081425}a^{15}+\frac{418}{1668934125}a^{13}+\frac{82343}{1001360475}a^{11}+\frac{14554}{333786825}a^{9}+\frac{2371}{22252455}a^{7}+\frac{22103}{7417485}a^{5}+\frac{24080}{164833}a^{3}+\frac{28445}{164833}a$, $\frac{1}{41\!\cdots\!75}a^{26}-\frac{2}{27036732825}a^{16}+\frac{71758}{45061221375}a^{14}-\frac{15743}{15020407125}a^{12}+\frac{25543}{333786825}a^{10}+\frac{583}{2472495}a^{8}+\frac{45214}{22252455}a^{6}+\frac{1379}{1483497}a^{4}+\frac{70847}{494499}a^{2}-\frac{235}{863}$, $\frac{1}{41\!\cdots\!75}a^{27}+\frac{10024}{9012244275}a^{15}-\frac{24263}{15020407125}a^{13}-\frac{42491}{1001360475}a^{11}-\frac{12571}{66757365}a^{9}-\frac{264}{824165}a^{7}-\frac{2842}{824165}a^{5}-\frac{56446}{494499}a^{3}-\frac{11727}{164833}a$, $\frac{1}{61\!\cdots\!25}a^{28}+\frac{26}{225306106875}a^{16}+\frac{51607}{45061221375}a^{14}-\frac{20717}{15020407125}a^{12}-\frac{65669}{1001360475}a^{10}-\frac{51631}{333786825}a^{8}-\frac{29449}{22252455}a^{6}-\frac{11621}{2472495}a^{4}+\frac{8045}{494499}a^{2}-\frac{423}{863}$, $\frac{1}{61\!\cdots\!25}a^{29}+\frac{19663}{15020407125}a^{15}-\frac{2717}{556311375}a^{13}+\frac{241}{4450491}a^{11}-\frac{28597}{333786825}a^{9}-\frac{2258}{2472495}a^{7}-\frac{2903}{494499}a^{5}+\frac{81349}{494499}a^{3}-\frac{40213}{164833}a$, $\frac{1}{18\!\cdots\!75}a^{30}+\frac{41}{675918320625}a^{16}-\frac{11666}{45061221375}a^{14}+\frac{50902}{15020407125}a^{12}-\frac{104}{200272095}a^{10}+\frac{17383}{333786825}a^{8}-\frac{15088}{7417485}a^{6}-\frac{1927}{1483497}a^{4}-\frac{5828}{164833}a^{2}-\frac{332}{863}$, $\frac{1}{18\!\cdots\!75}a^{31}-\frac{79636}{45061221375}a^{15}-\frac{73978}{15020407125}a^{13}-\frac{34708}{1001360475}a^{11}+\frac{67529}{333786825}a^{9}+\frac{16801}{7417485}a^{7}-\frac{52237}{7417485}a^{5}+\frac{50633}{494499}a^{3}-\frac{37855}{164833}a$ 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:  $15$
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 + 1226865178750781250)
 
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 + 1226865178750781250, 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 + 1226865178750781250);
 
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 + 1226865178750781250);
 
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 $C_{32}$ is not computed

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$