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Properties

Label 36.36.236...704.2
Degree $36$
Signature $[36, 0]$
Discriminant $2.361\times 10^{67}$
Root discriminant \(74.38\)
Ramified primes $2,3,7$
Class number not computed
Class group not computed
Galois group $C_6^2$ (as 36T4)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 72*x^34 + 2376*x^32 - 47608*x^30 + 646800*x^28 - 6301152*x^26 + 45393920*x^24 - 245794560*x^22 + 1006689024*x^20 - 3113398272*x^18 + 7203796992*x^16 - 12252653568*x^14 + 14896656384*x^12 - 12409675776*x^10 + 6646136832*x^8 - 2076114944*x^6 + 324796416*x^4 - 18874368*x^2 + 262144)
 
gp: K = bnfinit(y^36 - 72*y^34 + 2376*y^32 - 47608*y^30 + 646800*y^28 - 6301152*y^26 + 45393920*y^24 - 245794560*y^22 + 1006689024*y^20 - 3113398272*y^18 + 7203796992*y^16 - 12252653568*y^14 + 14896656384*y^12 - 12409675776*y^10 + 6646136832*y^8 - 2076114944*y^6 + 324796416*y^4 - 18874368*y^2 + 262144, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 72*x^34 + 2376*x^32 - 47608*x^30 + 646800*x^28 - 6301152*x^26 + 45393920*x^24 - 245794560*x^22 + 1006689024*x^20 - 3113398272*x^18 + 7203796992*x^16 - 12252653568*x^14 + 14896656384*x^12 - 12409675776*x^10 + 6646136832*x^8 - 2076114944*x^6 + 324796416*x^4 - 18874368*x^2 + 262144);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 72*x^34 + 2376*x^32 - 47608*x^30 + 646800*x^28 - 6301152*x^26 + 45393920*x^24 - 245794560*x^22 + 1006689024*x^20 - 3113398272*x^18 + 7203796992*x^16 - 12252653568*x^14 + 14896656384*x^12 - 12409675776*x^10 + 6646136832*x^8 - 2076114944*x^6 + 324796416*x^4 - 18874368*x^2 + 262144)
 

\( x^{36} - 72 x^{34} + 2376 x^{32} - 47608 x^{30} + 646800 x^{28} - 6301152 x^{26} + 45393920 x^{24} + \cdots + 262144 \) Copy content Toggle raw display

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

Invariants

Degree:  $36$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[36, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(23610692285332399309092778573219694177406932269512586359440570056704\) \(\medspace = 2^{54}\cdot 3^{54}\cdot 7^{30}\) 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:  \(74.38\)
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:  $2^{3/2}3^{3/2}7^{5/6}\approx 74.38326581667904$
Ramified primes:   \(2\), \(3\), \(7\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $36$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(504=2^{3}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{504}(1,·)$, $\chi_{504}(107,·)$, $\chi_{504}(257,·)$, $\chi_{504}(307,·)$, $\chi_{504}(11,·)$, $\chi_{504}(115,·)$, $\chi_{504}(17,·)$, $\chi_{504}(275,·)$, $\chi_{504}(25,·)$, $\chi_{504}(155,·)$, $\chi_{504}(289,·)$, $\chi_{504}(283,·)$, $\chi_{504}(41,·)$, $\chi_{504}(179,·)$, $\chi_{504}(185,·)$, $\chi_{504}(347,·)$, $\chi_{504}(187,·)$, $\chi_{504}(193,·)$, $\chi_{504}(139,·)$, $\chi_{504}(457,·)$, $\chi_{504}(337,·)$, $\chi_{504}(377,·)$, $\chi_{504}(89,·)$, $\chi_{504}(475,·)$, $\chi_{504}(353,·)$, $\chi_{504}(355,·)$, $\chi_{504}(425,·)$, $\chi_{504}(209,·)$, $\chi_{504}(361,·)$, $\chi_{504}(323,·)$, $\chi_{504}(491,·)$, $\chi_{504}(451,·)$, $\chi_{504}(19,·)$, $\chi_{504}(169,·)$, $\chi_{504}(121,·)$, $\chi_{504}(443,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$, $\frac{1}{4096}a^{24}$, $\frac{1}{4096}a^{25}$, $\frac{1}{8192}a^{26}$, $\frac{1}{8192}a^{27}$, $\frac{1}{16384}a^{28}$, $\frac{1}{16384}a^{29}$, $\frac{1}{32768}a^{30}$, $\frac{1}{32768}a^{31}$, $\frac{1}{65536}a^{32}$, $\frac{1}{65536}a^{33}$, $\frac{1}{131072}a^{34}$, $\frac{1}{131072}a^{35}$ 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:  $35$
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^36 - 72*x^34 + 2376*x^32 - 47608*x^30 + 646800*x^28 - 6301152*x^26 + 45393920*x^24 - 245794560*x^22 + 1006689024*x^20 - 3113398272*x^18 + 7203796992*x^16 - 12252653568*x^14 + 14896656384*x^12 - 12409675776*x^10 + 6646136832*x^8 - 2076114944*x^6 + 324796416*x^4 - 18874368*x^2 + 262144)
 
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^36 - 72*x^34 + 2376*x^32 - 47608*x^30 + 646800*x^28 - 6301152*x^26 + 45393920*x^24 - 245794560*x^22 + 1006689024*x^20 - 3113398272*x^18 + 7203796992*x^16 - 12252653568*x^14 + 14896656384*x^12 - 12409675776*x^10 + 6646136832*x^8 - 2076114944*x^6 + 324796416*x^4 - 18874368*x^2 + 262144, 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^36 - 72*x^34 + 2376*x^32 - 47608*x^30 + 646800*x^28 - 6301152*x^26 + 45393920*x^24 - 245794560*x^22 + 1006689024*x^20 - 3113398272*x^18 + 7203796992*x^16 - 12252653568*x^14 + 14896656384*x^12 - 12409675776*x^10 + 6646136832*x^8 - 2076114944*x^6 + 324796416*x^4 - 18874368*x^2 + 262144);
 
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^36 - 72*x^34 + 2376*x^32 - 47608*x^30 + 646800*x^28 - 6301152*x^26 + 45393920*x^24 - 245794560*x^22 + 1006689024*x^20 - 3113398272*x^18 + 7203796992*x^16 - 12252653568*x^14 + 14896656384*x^12 - 12409675776*x^10 + 6646136832*x^8 - 2076114944*x^6 + 324796416*x^4 - 18874368*x^2 + 262144);
 
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

$C_6^2$ (as 36T4):

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 36
The 36 conjugacy class representatives for $C_6^2$
Character table for $C_6^2$ is not computed

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{6}, \sqrt{14})\), 6.6.10077696.1, 6.6.24196548096.1, 6.6.24196548096.2, 6.6.33191424.1, 6.6.1152216576.1, 6.6.6751269.1, 6.6.56458612224.1, 6.6.330812181.2, 6.6.56458612224.2, 6.6.330812181.1, 6.6.8605184.1, \(\Q(\zeta_{21})^+\), 9.9.62523502209.1, 12.12.11948427342082473984.1, 12.12.28688174048340020035584.1, 12.12.28688174048340020035584.3, 12.12.53981860730241024.2, 18.18.14166424145858741301073711988736.1, 18.18.179966054889983269121047526375424.1, \(\Q(\zeta_{63})^+\)

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 ${\href{/padicField/5.3.0.1}{3} }^{12}$ R ${\href{/padicField/11.6.0.1}{6} }^{6}$ ${\href{/padicField/13.6.0.1}{6} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{6}$ ${\href{/padicField/19.6.0.1}{6} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{6}$ ${\href{/padicField/29.6.0.1}{6} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{6}$ ${\href{/padicField/41.6.0.1}{6} }^{6}$ ${\href{/padicField/43.3.0.1}{3} }^{12}$ ${\href{/padicField/47.3.0.1}{3} }^{12}$ ${\href{/padicField/53.6.0.1}{6} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{6}$

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 2.12.18.15$x^{12} - 12 x^{11} - 56 x^{9} + 2372 x^{8} + 8992 x^{7} + 115648 x^{6} + 164160 x^{5} + 1305648 x^{4} - 282560 x^{3} + 5591296 x^{2} - 9640320 x + 8497856$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
2.12.18.15$x^{12} - 12 x^{11} - 56 x^{9} + 2372 x^{8} + 8992 x^{7} + 115648 x^{6} + 164160 x^{5} + 1305648 x^{4} - 282560 x^{3} + 5591296 x^{2} - 9640320 x + 8497856$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
2.12.18.15$x^{12} - 12 x^{11} - 56 x^{9} + 2372 x^{8} + 8992 x^{7} + 115648 x^{6} + 164160 x^{5} + 1305648 x^{4} - 282560 x^{3} + 5591296 x^{2} - 9640320 x + 8497856$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
\(3\) Copy content Toggle raw display Deg $36$$6$$6$$54$
\(7\) Copy content Toggle raw display Deg $36$$6$$6$$30$