Properties

Label 36.0.236...704.3
Degree $36$
Signature $[0, 18]$
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)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 12*x^34 + 108*x^32 - 888*x^30 + 7056*x^28 - 55296*x^26 + 430848*x^24 - 1651968*x^22 + 5640192*x^20 - 18551808*x^18 + 59222016*x^16 - 178163712*x^14 + 451215360*x^12 - 422019072*x^10 + 394149888*x^8 - 366280704*x^6 + 334430208*x^4 - 286654464*x^2 + 191102976)
 
gp: K = bnfinit(y^36 - 12*y^34 + 108*y^32 - 888*y^30 + 7056*y^28 - 55296*y^26 + 430848*y^24 - 1651968*y^22 + 5640192*y^20 - 18551808*y^18 + 59222016*y^16 - 178163712*y^14 + 451215360*y^12 - 422019072*y^10 + 394149888*y^8 - 366280704*y^6 + 334430208*y^4 - 286654464*y^2 + 191102976, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 12*x^34 + 108*x^32 - 888*x^30 + 7056*x^28 - 55296*x^26 + 430848*x^24 - 1651968*x^22 + 5640192*x^20 - 18551808*x^18 + 59222016*x^16 - 178163712*x^14 + 451215360*x^12 - 422019072*x^10 + 394149888*x^8 - 366280704*x^6 + 334430208*x^4 - 286654464*x^2 + 191102976);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 12*x^34 + 108*x^32 - 888*x^30 + 7056*x^28 - 55296*x^26 + 430848*x^24 - 1651968*x^22 + 5640192*x^20 - 18551808*x^18 + 59222016*x^16 - 178163712*x^14 + 451215360*x^12 - 422019072*x^10 + 394149888*x^8 - 366280704*x^6 + 334430208*x^4 - 286654464*x^2 + 191102976)
 

\( x^{36} - 12 x^{34} + 108 x^{32} - 888 x^{30} + 7056 x^{28} - 55296 x^{26} + 430848 x^{24} - 1651968 x^{22} + 5640192 x^{20} - 18551808 x^{18} + 59222016 x^{16} + \cdots + 191102976 \) 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:  $[0, 18]$
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$$\chi_{504}(1,·)$, $\chi_{504}(5,·)$, $\chi_{504}(265,·)$, $\chi_{504}(269,·)$, $\chi_{504}(145,·)$, $\chi_{504}(149,·)$, $\chi_{504}(25,·)$, $\chi_{504}(409,·)$, $\chi_{504}(29,·)$, $\chi_{504}(389,·)$, $\chi_{504}(289,·)$, $\chi_{504}(293,·)$, $\chi_{504}(169,·)$, $\chi_{504}(173,·)$, $\chi_{504}(433,·)$, $\chi_{504}(53,·)$, $\chi_{504}(73,·)$, $\chi_{504}(313,·)$, $\chi_{504}(317,·)$, $\chi_{504}(437,·)$, $\chi_{504}(193,·)$, $\chi_{504}(197,·)$, $\chi_{504}(97,·)$, $\chi_{504}(457,·)$, $\chi_{504}(461,·)$, $\chi_{504}(337,·)$, $\chi_{504}(341,·)$, $\chi_{504}(221,·)$, $\chi_{504}(101,·)$, $\chi_{504}(481,·)$, $\chi_{504}(485,·)$, $\chi_{504}(361,·)$, $\chi_{504}(365,·)$, $\chi_{504}(241,·)$, $\chi_{504}(121,·)$, $\chi_{504}(125,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{131072}$

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

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{24}a^{6}$, $\frac{1}{24}a^{7}$, $\frac{1}{48}a^{8}$, $\frac{1}{48}a^{9}$, $\frac{1}{96}a^{10}$, $\frac{1}{96}a^{11}$, $\frac{1}{576}a^{12}$, $\frac{1}{576}a^{13}$, $\frac{1}{4608}a^{14}-\frac{1}{4}$, $\frac{1}{4608}a^{15}-\frac{1}{4}a$, $\frac{1}{9216}a^{16}-\frac{1}{8}a^{2}$, $\frac{1}{9216}a^{17}-\frac{1}{8}a^{3}$, $\frac{1}{55296}a^{18}+\frac{1}{16}a^{4}$, $\frac{1}{55296}a^{19}+\frac{1}{16}a^{5}$, $\frac{1}{110592}a^{20}-\frac{1}{96}a^{6}$, $\frac{1}{110592}a^{21}-\frac{1}{96}a^{7}$, $\frac{1}{221184}a^{22}-\frac{1}{192}a^{8}$, $\frac{1}{221184}a^{23}-\frac{1}{192}a^{9}$, $\frac{1}{1327104}a^{24}+\frac{1}{384}a^{10}$, $\frac{1}{1327104}a^{25}+\frac{1}{384}a^{11}$, $\frac{1}{37124407296}a^{26}-\frac{3353}{9281101824}a^{24}+\frac{901}{515616768}a^{22}+\frac{139}{128904192}a^{20}+\frac{499}{96678144}a^{18}-\frac{1469}{42968064}a^{16}-\frac{1189}{64452096}a^{14}+\frac{22139}{32226048}a^{12}+\frac{3643}{2685504}a^{10}+\frac{1777}{447584}a^{8}+\frac{181}{111896}a^{6}+\frac{3083}{55948}a^{4}+\frac{6739}{111896}a^{2}-\frac{27711}{55948}$, $\frac{1}{37124407296}a^{27}-\frac{3353}{9281101824}a^{25}+\frac{901}{515616768}a^{23}+\frac{139}{128904192}a^{21}+\frac{499}{96678144}a^{19}-\frac{1469}{42968064}a^{17}-\frac{1189}{64452096}a^{15}+\frac{22139}{32226048}a^{13}+\frac{3643}{2685504}a^{11}+\frac{1777}{447584}a^{9}+\frac{181}{111896}a^{7}+\frac{3083}{55948}a^{5}+\frac{6739}{111896}a^{3}-\frac{27711}{55948}a$, $\frac{1}{296995258368}a^{28}-\frac{10333}{128904192}a^{14}+\frac{99225}{223792}$, $\frac{1}{296995258368}a^{29}-\frac{10333}{128904192}a^{15}+\frac{99225}{223792}a$, $\frac{1}{1781971550208}a^{30}-\frac{12769}{257808384}a^{16}-\frac{22873}{447584}a^{2}$, $\frac{1}{1781971550208}a^{31}-\frac{12769}{257808384}a^{17}-\frac{22873}{447584}a^{3}$, $\frac{1}{3563943100416}a^{32}-\frac{10333}{1546850304}a^{18}+\frac{33075}{895168}a^{4}$, $\frac{1}{3563943100416}a^{33}-\frac{10333}{1546850304}a^{19}+\frac{33075}{895168}a^{5}$, $\frac{1}{7127886200832}a^{34}-\frac{10333}{3093700608}a^{20}+\frac{33075}{1790336}a^{6}$, $\frac{1}{7127886200832}a^{35}-\frac{10333}{3093700608}a^{21}+\frac{33075}{1790336}a^{7}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

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:  $17$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{1165}{7127886200832} a^{34} + \frac{95341}{343744512} a^{20} + \frac{6554473}{5371008} a^{6} \)  (order $14$) 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 - 12*x^34 + 108*x^32 - 888*x^30 + 7056*x^28 - 55296*x^26 + 430848*x^24 - 1651968*x^22 + 5640192*x^20 - 18551808*x^18 + 59222016*x^16 - 178163712*x^14 + 451215360*x^12 - 422019072*x^10 + 394149888*x^8 - 366280704*x^6 + 334430208*x^4 - 286654464*x^2 + 191102976)
 
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 - 12*x^34 + 108*x^32 - 888*x^30 + 7056*x^28 - 55296*x^26 + 430848*x^24 - 1651968*x^22 + 5640192*x^20 - 18551808*x^18 + 59222016*x^16 - 178163712*x^14 + 451215360*x^12 - 422019072*x^10 + 394149888*x^8 - 366280704*x^6 + 334430208*x^4 - 286654464*x^2 + 191102976, 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 - 12*x^34 + 108*x^32 - 888*x^30 + 7056*x^28 - 55296*x^26 + 430848*x^24 - 1651968*x^22 + 5640192*x^20 - 18551808*x^18 + 59222016*x^16 - 178163712*x^14 + 451215360*x^12 - 422019072*x^10 + 394149888*x^8 - 366280704*x^6 + 334430208*x^4 - 286654464*x^2 + 191102976);
 
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 - 12*x^34 + 108*x^32 - 888*x^30 + 7056*x^28 - 55296*x^26 + 430848*x^24 - 1651968*x^22 + 5640192*x^20 - 18551808*x^18 + 59222016*x^16 - 178163712*x^14 + 451215360*x^12 - 422019072*x^10 + 394149888*x^8 - 366280704*x^6 + 334430208*x^4 - 286654464*x^2 + 191102976);
 
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

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{42}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\sqrt{-6}, \sqrt{-7})\), 6.0.10077696.1, 6.0.33191424.1, 6.0.24196548096.1, 6.0.24196548096.2, 6.0.2250423.1, 6.6.3456649728.1, \(\Q(\zeta_{7})\), 6.6.232339968.1, 6.0.110270727.2, 6.6.169375836672.1, 6.0.110270727.1, 6.6.169375836672.2, 9.9.62523502209.1, 12.0.11948427342082473984.3, 12.0.53981860730241024.5, 12.0.28688174048340020035584.6, 12.0.28688174048340020035584.8, 18.0.14166424145858741301073711988736.4, 18.0.1340851596668237962730583.1, 18.18.4859083482029548266268283212136448.1

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 ${\href{/padicField/5.6.0.1}{6} }^{6}$ R ${\href{/padicField/11.3.0.1}{3} }^{12}$ ${\href{/padicField/13.6.0.1}{6} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{6}$ ${\href{/padicField/19.6.0.1}{6} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{6}$ ${\href{/padicField/29.3.0.1}{3} }^{12}$ ${\href{/padicField/31.6.0.1}{6} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{6}$ ${\href{/padicField/41.6.0.1}{6} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{6}$ ${\href{/padicField/53.3.0.1}{3} }^{12}$ ${\href{/padicField/59.6.0.1}{6} }^{6}$

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 2.6.9.3$x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$$2$$3$$9$$C_6$$[3]^{3}$
\(3\) Copy content Toggle raw display Deg $36$$6$$6$$54$
\(7\) Copy content Toggle raw display 7.18.15.5$x^{18} + 36 x^{17} + 540 x^{16} + 4344 x^{15} + 20160 x^{14} + 55296 x^{13} + 98757 x^{12} + 161784 x^{11} + 246024 x^{10} + 264920 x^{9} + 530640 x^{8} + 156384 x^{7} - 1885725 x^{6} - 6133212 x^{5} - 3645540 x^{4} + 5968464 x^{3} + 5011344 x^{2} + 1820448 x + 2358791$$6$$3$$15$$C_6 \times C_3$$[\ ]_{6}^{3}$
7.18.15.5$x^{18} + 36 x^{17} + 540 x^{16} + 4344 x^{15} + 20160 x^{14} + 55296 x^{13} + 98757 x^{12} + 161784 x^{11} + 246024 x^{10} + 264920 x^{9} + 530640 x^{8} + 156384 x^{7} - 1885725 x^{6} - 6133212 x^{5} - 3645540 x^{4} + 5968464 x^{3} + 5011344 x^{2} + 1820448 x + 2358791$$6$$3$$15$$C_6 \times C_3$$[\ ]_{6}^{3}$