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Properties

Label 32.0.203...000.7
Degree $32$
Signature $[0, 16]$
Discriminant $2.033\times 10^{53}$
Root discriminant \(46.33\)
Ramified primes $2,3,5$
Class number $3200$ (GRH)
Class group [40, 80] (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 32*x^30 + 464*x^28 + 4032*x^26 + 23401*x^24 + 95704*x^22 + 283612*x^20 + 616816*x^18 + 986442*x^16 + 1151648*x^14 + 965328*x^12 + 564928*x^10 + 220856*x^8 + 53696*x^6 + 7136*x^4 + 384*x^2 + 1)
 
gp: K = bnfinit(y^32 + 32*y^30 + 464*y^28 + 4032*y^26 + 23401*y^24 + 95704*y^22 + 283612*y^20 + 616816*y^18 + 986442*y^16 + 1151648*y^14 + 965328*y^12 + 564928*y^10 + 220856*y^8 + 53696*y^6 + 7136*y^4 + 384*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 32*x^30 + 464*x^28 + 4032*x^26 + 23401*x^24 + 95704*x^22 + 283612*x^20 + 616816*x^18 + 986442*x^16 + 1151648*x^14 + 965328*x^12 + 564928*x^10 + 220856*x^8 + 53696*x^6 + 7136*x^4 + 384*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + 32*x^30 + 464*x^28 + 4032*x^26 + 23401*x^24 + 95704*x^22 + 283612*x^20 + 616816*x^18 + 986442*x^16 + 1151648*x^14 + 965328*x^12 + 564928*x^10 + 220856*x^8 + 53696*x^6 + 7136*x^4 + 384*x^2 + 1)
 

\( x^{32} + 32 x^{30} + 464 x^{28} + 4032 x^{26} + 23401 x^{24} + 95704 x^{22} + 283612 x^{20} + 616816 x^{18} + \cdots + 1 \) 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:   \(203282392447840896882957090816000000000000000000000000\) \(\medspace = 2^{96}\cdot 3^{16}\cdot 5^{24}\) 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:  \(46.33\)
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:  $2^{3}3^{1/2}5^{3/4}\approx 46.331687411530766$
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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $32$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(240=2^{4}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{240}(1,·)$, $\chi_{240}(133,·)$, $\chi_{240}(7,·)$, $\chi_{240}(137,·)$, $\chi_{240}(139,·)$, $\chi_{240}(13,·)$, $\chi_{240}(17,·)$, $\chi_{240}(19,·)$, $\chi_{240}(149,·)$, $\chi_{240}(157,·)$, $\chi_{240}(37,·)$, $\chi_{240}(169,·)$, $\chi_{240}(29,·)$, $\chi_{240}(49,·)$, $\chi_{240}(191,·)$, $\chi_{240}(71,·)$, $\chi_{240}(203,·)$, $\chi_{240}(83,·)$, $\chi_{240}(91,·)$, $\chi_{240}(221,·)$, $\chi_{240}(223,·)$, $\chi_{240}(227,·)$, $\chi_{240}(101,·)$, $\chi_{240}(103,·)$, $\chi_{240}(233,·)$, $\chi_{240}(107,·)$, $\chi_{240}(239,·)$, $\chi_{240}(113,·)$, $\chi_{240}(211,·)$, $\chi_{240}(119,·)$, $\chi_{240}(121,·)$, $\chi_{240}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$ Copy content Toggle raw display

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

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{40}\times C_{80}$, which has order $3200$ (assuming GRH)

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:   $a^{24}+24a^{22}+252a^{20}+1520a^{18}+5814a^{16}+14688a^{14}+24752a^{12}+27456a^{10}+19305a^{8}+8008a^{6}+1716a^{4}+144a^{2}+1$, $2a^{28}+56a^{26}+700a^{24}+5152a^{22}+24795a^{20}+81948a^{18}+190094a^{16}+310880a^{14}+354990a^{12}+276264a^{10}+140372a^{8}+43296a^{6}+7089a^{4}+452a^{2}+2$, $a^{28}+28a^{26}+350a^{24}+2576a^{22}+12397a^{20}+40964a^{18}+94962a^{16}+155040a^{14}+176357a^{12}+136124a^{10}+68014a^{8}+20272a^{6}+3080a^{4}+160a^{2}$, $a^{18}+18a^{16}+135a^{14}+546a^{12}+1287a^{10}+1782a^{8}+1386a^{6}+540a^{4}+81a^{2}+2$, $a^{6}+6a^{4}+9a^{2}+2$, $a^{30}+30a^{28}+406a^{26}+3276a^{24}+17549a^{22}+65758a^{20}+176891a^{18}+344982a^{16}+486572a^{14}+489384a^{12}+341550a^{10}+158092a^{8}+45045a^{6}+6910a^{4}+425a^{2}+2$, $a^{28}+28a^{26}+350a^{24}+2576a^{22}+12397a^{20}+40964a^{18}+94962a^{16}+155040a^{14}+176358a^{12}+136136a^{10}+68068a^{8}+20384a^{6}+3185a^{4}+196a^{2}+3$, $a^{31}+31a^{29}+434a^{27}+3626a^{25}+20125a^{23}+78155a^{21}+217854a^{19}+439925a^{17}+641460a^{15}+665076a^{13}+475944a^{11}+223378a^{9}+62764a^{7}+8652a^{5}+232a^{3}-32a$, $a^{3}+3a$, $a^{29}+29a^{27}+377a^{25}+2900a^{23}+14674a^{21}+51359a^{19}+127281a^{17}+224808a^{15}+281010a^{13}+243543a^{11}+141009a^{9}+51316a^{7}+10633a^{5}+1070a^{3}+40a$, $a^{27}+27a^{25}+324a^{23}+2277a^{21}+10395a^{19}+32319a^{17}+69768a^{15}+104652a^{13}+107406a^{11}+72930a^{9}+30888a^{7}+7371a^{5}+819a^{3}+27a$, $a^{5}+5a^{3}+5a$, $a^{21}+22a^{19}+208a^{17}+1104a^{15}+3605a^{13}+7462a^{11}+9724a^{9}+7656a^{7}+3333a^{5}+670a^{3}+40a$, $a^{17}+17a^{15}+119a^{13}+442a^{11}+935a^{9}+1122a^{7}+714a^{5}+204a^{3}+17a$, $a^{27}+27a^{25}+324a^{23}+2277a^{21}+10395a^{19}+32319a^{17}+69768a^{15}+104653a^{13}+107419a^{11}+72995a^{9}+31044a^{7}+7553a^{5}+910a^{3}+40a$ Copy content Toggle raw display (assuming GRH)
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:  \( 5926511257.21094 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 5926511257.21094 \cdot 3200}{2\cdot\sqrt{203282392447840896882957090816000000000000000000000000}}\cr\approx \mathstrut & 0.124092949839740 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 + 32*x^30 + 464*x^28 + 4032*x^26 + 23401*x^24 + 95704*x^22 + 283612*x^20 + 616816*x^18 + 986442*x^16 + 1151648*x^14 + 965328*x^12 + 564928*x^10 + 220856*x^8 + 53696*x^6 + 7136*x^4 + 384*x^2 + 1)
 
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 + 32*x^30 + 464*x^28 + 4032*x^26 + 23401*x^24 + 95704*x^22 + 283612*x^20 + 616816*x^18 + 986442*x^16 + 1151648*x^14 + 965328*x^12 + 564928*x^10 + 220856*x^8 + 53696*x^6 + 7136*x^4 + 384*x^2 + 1, 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 + 32*x^30 + 464*x^28 + 4032*x^26 + 23401*x^24 + 95704*x^22 + 283612*x^20 + 616816*x^18 + 986442*x^16 + 1151648*x^14 + 965328*x^12 + 564928*x^10 + 220856*x^8 + 53696*x^6 + 7136*x^4 + 384*x^2 + 1);
 
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 + 32*x^30 + 464*x^28 + 4032*x^26 + 23401*x^24 + 95704*x^22 + 283612*x^20 + 616816*x^18 + 986442*x^16 + 1151648*x^14 + 965328*x^12 + 564928*x^10 + 220856*x^8 + 53696*x^6 + 7136*x^4 + 384*x^2 + 1);
 
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

$C_2\times C_4^2$ (as 32T36):

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)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{2}, \sqrt{3})\), 4.0.18432.2, 4.0.2048.2, \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{15})\), 4.0.460800.2, 4.0.51200.2, \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{5})\), 4.0.256000.4, 4.0.2304000.2, 4.0.256000.2, 4.0.2304000.1, \(\Q(\zeta_{15})^+\), 4.4.8000.1, 4.4.72000.1, \(\Q(\zeta_{20})^+\), 8.0.1358954496.3, 8.8.3317760000.1, 8.0.849346560000.4, 8.0.212336640000.5, 8.0.849346560000.1, 8.0.2621440000.1, 8.0.849346560000.2, 8.0.21233664000000.3, 8.0.21233664000000.1, 8.8.5184000000.2, 8.8.82944000000.2, 8.0.65536000000.1, 8.0.5308416000000.7, 8.8.5184000000.1, \(\Q(\zeta_{40})^+\), 8.0.21233664000000.4, 8.0.21233664000000.2, \(\Q(\zeta_{60})^+\), 8.8.82944000000.1, 16.0.721389578983833600000000.4, 16.0.450868486864896000000000000.4, \(\Q(\zeta_{120})^+\), 16.0.28179280429056000000000000.4, 16.0.450868486864896000000000000.7, 16.0.68719476736000000000000.2, 16.0.28179280429056000000000000.6

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 ${\href{/padicField/7.4.0.1}{4} }^{8}$ ${\href{/padicField/11.4.0.1}{4} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{16}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{8}$

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$$8$$4$$96$
\(3\) Copy content Toggle raw display 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(5\) Copy content Toggle raw display 5.16.12.1$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$
5.16.12.1$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$