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Properties

Label 36.0.499...625.3
Degree $36$
Signature $[0, 18]$
Discriminant $5.000\times 10^{63}$
Root discriminant \(58.81\)
Ramified primes $3,5,7$
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 - x^33 - 70*x^30 + 1289*x^27 + 2848*x^24 - 33978*x^21 + 415031*x^18 - 2654421*x^15 + 8225230*x^12 + 50229628*x^9 + 90247619*x^6 + 275010364*x^3 + 594823321)
 
gp: K = bnfinit(y^36 - y^33 - 70*y^30 + 1289*y^27 + 2848*y^24 - 33978*y^21 + 415031*y^18 - 2654421*y^15 + 8225230*y^12 + 50229628*y^9 + 90247619*y^6 + 275010364*y^3 + 594823321, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - x^33 - 70*x^30 + 1289*x^27 + 2848*x^24 - 33978*x^21 + 415031*x^18 - 2654421*x^15 + 8225230*x^12 + 50229628*x^9 + 90247619*x^6 + 275010364*x^3 + 594823321);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - x^33 - 70*x^30 + 1289*x^27 + 2848*x^24 - 33978*x^21 + 415031*x^18 - 2654421*x^15 + 8225230*x^12 + 50229628*x^9 + 90247619*x^6 + 275010364*x^3 + 594823321)
 

\( x^{36} - x^{33} - 70 x^{30} + 1289 x^{27} + 2848 x^{24} - 33978 x^{21} + 415031 x^{18} + \cdots + 594823321 \) 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:   \(4999758568289868528789868885747458073284974388119052886962890625\) \(\medspace = 3^{54}\cdot 5^{18}\cdot 7^{30}\) 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:  \(58.81\)
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:  $3^{3/2}5^{1/2}7^{5/6}\approx 58.8051349456126$
Ramified primes:   \(3\), \(5\), \(7\) 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:  \(315=3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{315}(256,·)$, $\chi_{315}(1,·)$, $\chi_{315}(139,·)$, $\chi_{315}(269,·)$, $\chi_{315}(16,·)$, $\chi_{315}(19,·)$, $\chi_{315}(151,·)$, $\chi_{315}(281,·)$, $\chi_{315}(304,·)$, $\chi_{315}(34,·)$, $\chi_{315}(164,·)$, $\chi_{315}(296,·)$, $\chi_{315}(299,·)$, $\chi_{315}(46,·)$, $\chi_{315}(176,·)$, $\chi_{315}(116,·)$, $\chi_{315}(314,·)$, $\chi_{315}(59,·)$, $\chi_{315}(191,·)$, $\chi_{315}(194,·)$, $\chi_{315}(11,·)$, $\chi_{315}(199,·)$, $\chi_{315}(209,·)$, $\chi_{315}(211,·)$, $\chi_{315}(86,·)$, $\chi_{315}(89,·)$, $\chi_{315}(71,·)$, $\chi_{315}(221,·)$, $\chi_{315}(94,·)$, $\chi_{315}(226,·)$, $\chi_{315}(229,·)$, $\chi_{315}(104,·)$, $\chi_{315}(106,·)$, $\chi_{315}(244,·)$, $\chi_{315}(121,·)$, $\chi_{315}(124,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{13}a^{18}+\frac{5}{13}a^{15}+\frac{5}{13}a^{9}+\frac{5}{13}a^{3}-\frac{1}{13}$, $\frac{1}{13}a^{19}+\frac{5}{13}a^{16}+\frac{5}{13}a^{10}+\frac{5}{13}a^{4}-\frac{1}{13}a$, $\frac{1}{13}a^{20}+\frac{5}{13}a^{17}+\frac{5}{13}a^{11}+\frac{5}{13}a^{5}-\frac{1}{13}a^{2}$, $\frac{1}{13}a^{21}+\frac{1}{13}a^{15}+\frac{5}{13}a^{12}+\frac{1}{13}a^{9}+\frac{5}{13}a^{6}+\frac{5}{13}$, $\frac{1}{13}a^{22}+\frac{1}{13}a^{16}+\frac{5}{13}a^{13}+\frac{1}{13}a^{10}+\frac{5}{13}a^{7}+\frac{5}{13}a$, $\frac{1}{13}a^{23}+\frac{1}{13}a^{17}+\frac{5}{13}a^{14}+\frac{1}{13}a^{11}+\frac{5}{13}a^{8}+\frac{5}{13}a^{2}$, $\frac{1}{13}a^{24}+\frac{1}{13}a^{12}+\frac{1}{13}$, $\frac{1}{13}a^{25}+\frac{1}{13}a^{13}+\frac{1}{13}a$, $\frac{1}{13}a^{26}+\frac{1}{13}a^{14}+\frac{1}{13}a^{2}$, $\frac{1}{26}a^{27}-\frac{6}{13}a^{15}-\frac{1}{2}a^{9}-\frac{6}{13}a^{3}-\frac{1}{2}$, $\frac{1}{26}a^{28}-\frac{6}{13}a^{16}-\frac{1}{2}a^{10}-\frac{6}{13}a^{4}-\frac{1}{2}a$, $\frac{1}{26}a^{29}-\frac{6}{13}a^{17}-\frac{1}{2}a^{11}-\frac{6}{13}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4112401046}a^{30}-\frac{25393591}{4112401046}a^{27}+\frac{12696760}{2056200523}a^{24}+\frac{14910783}{2056200523}a^{21}-\frac{31936432}{2056200523}a^{18}+\frac{454998186}{2056200523}a^{15}-\frac{1362198399}{4112401046}a^{12}+\frac{1864147897}{4112401046}a^{9}-\frac{722929869}{2056200523}a^{6}-\frac{889789065}{4112401046}a^{3}+\frac{484278341}{4112401046}$, $\frac{1}{119259630334}a^{31}-\frac{408119973}{59629815167}a^{28}-\frac{1885334492}{59629815167}a^{25}+\frac{173080054}{59629815167}a^{22}+\frac{442571381}{59629815167}a^{19}+\frac{1562183083}{59629815167}a^{16}+\frac{16985437037}{119259630334}a^{13}+\frac{22996687253}{59629815167}a^{10}-\frac{16381687698}{59629815167}a^{7}-\frac{44228169319}{119259630334}a^{4}-\frac{12648656416}{59629815167}a$, $\frac{1}{3458529279686}a^{32}+\frac{3770668913}{3458529279686}a^{29}-\frac{1885334492}{1729264639843}a^{26}+\frac{9346897772}{1729264639843}a^{23}-\frac{54600334927}{1729264639843}a^{20}-\frac{649778874895}{1729264639843}a^{17}+\frac{1062800656889}{3458529279686}a^{14}-\frac{426458237971}{3458529279686}a^{11}+\frac{89117216059}{1729264639843}a^{8}+\frac{1496973207305}{3458529279686}a^{5}+\frac{951714274135}{3458529279686}a^{2}$, $\frac{1}{69\!\cdots\!06}a^{33}+\frac{51\!\cdots\!52}{34\!\cdots\!03}a^{30}-\frac{86\!\cdots\!25}{69\!\cdots\!06}a^{27}+\frac{66\!\cdots\!22}{34\!\cdots\!03}a^{24}+\frac{74\!\cdots\!44}{34\!\cdots\!03}a^{21}-\frac{41\!\cdots\!05}{34\!\cdots\!03}a^{18}-\frac{13\!\cdots\!87}{69\!\cdots\!06}a^{15}-\frac{80\!\cdots\!43}{34\!\cdots\!03}a^{12}-\frac{34\!\cdots\!11}{69\!\cdots\!06}a^{9}-\frac{84\!\cdots\!97}{69\!\cdots\!06}a^{6}+\frac{11\!\cdots\!49}{34\!\cdots\!03}a^{3}-\frac{12\!\cdots\!53}{28\!\cdots\!54}$, $\frac{1}{20\!\cdots\!74}a^{34}+\frac{51\!\cdots\!52}{10\!\cdots\!87}a^{31}-\frac{98\!\cdots\!71}{10\!\cdots\!87}a^{28}+\frac{81\!\cdots\!15}{10\!\cdots\!87}a^{25}+\frac{22\!\cdots\!92}{10\!\cdots\!87}a^{22}+\frac{23\!\cdots\!74}{10\!\cdots\!87}a^{19}+\frac{28\!\cdots\!85}{20\!\cdots\!74}a^{16}-\frac{20\!\cdots\!37}{10\!\cdots\!87}a^{13}-\frac{48\!\cdots\!15}{10\!\cdots\!87}a^{10}+\frac{27\!\cdots\!89}{20\!\cdots\!74}a^{7}-\frac{13\!\cdots\!87}{10\!\cdots\!87}a^{4}+\frac{23\!\cdots\!46}{41\!\cdots\!83}a$, $\frac{1}{58\!\cdots\!46}a^{35}+\frac{51\!\cdots\!52}{29\!\cdots\!23}a^{32}+\frac{83\!\cdots\!47}{58\!\cdots\!46}a^{29}+\frac{24\!\cdots\!12}{29\!\cdots\!23}a^{26}+\frac{56\!\cdots\!85}{29\!\cdots\!23}a^{23}+\frac{11\!\cdots\!60}{29\!\cdots\!23}a^{20}-\frac{18\!\cdots\!77}{58\!\cdots\!46}a^{17}-\frac{12\!\cdots\!23}{29\!\cdots\!23}a^{14}-\frac{14\!\cdots\!59}{58\!\cdots\!46}a^{11}-\frac{14\!\cdots\!21}{58\!\cdots\!46}a^{8}+\frac{13\!\cdots\!40}{29\!\cdots\!23}a^{5}-\frac{54\!\cdots\!51}{24\!\cdots\!14}a^{2}$ 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:  $17$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{6921108753981151158125696013}{8292087143854494049769968630493817166} a^{34} - \frac{941567415194979960768417862}{142967019721629207754654631560238227} a^{31} - \frac{150652800368259247394568166143}{4146043571927247024884984315246908583} a^{28} + \frac{6095073959946102864921771660252}{4146043571927247024884984315246908583} a^{25} - \frac{26254952032201136127001100887891}{4146043571927247024884984315246908583} a^{22} - \frac{105340678329434777534244437457400}{4146043571927247024884984315246908583} a^{19} + \frac{5039082461939296081592750135788093}{8292087143854494049769968630493817166} a^{16} - \frac{21832931462959167005202935547890502}{4146043571927247024884984315246908583} a^{13} + \frac{111571895622143247251603156194403821}{4146043571927247024884984315246908583} a^{10} - \frac{374570198742050087237942648772652019}{8292087143854494049769968630493817166} a^{7} - \frac{18185607450678345871428102267198351}{142967019721629207754654631560238227} a^{4} + \frac{2347059412181283559934910054110578416}{4146043571927247024884984315246908583} a \)  (order $18$) 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 - x^33 - 70*x^30 + 1289*x^27 + 2848*x^24 - 33978*x^21 + 415031*x^18 - 2654421*x^15 + 8225230*x^12 + 50229628*x^9 + 90247619*x^6 + 275010364*x^3 + 594823321)
 
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 - x^33 - 70*x^30 + 1289*x^27 + 2848*x^24 - 33978*x^21 + 415031*x^18 - 2654421*x^15 + 8225230*x^12 + 50229628*x^9 + 90247619*x^6 + 275010364*x^3 + 594823321, 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 - x^33 - 70*x^30 + 1289*x^27 + 2848*x^24 - 33978*x^21 + 415031*x^18 - 2654421*x^15 + 8225230*x^12 + 50229628*x^9 + 90247619*x^6 + 275010364*x^3 + 594823321);
 
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 - x^33 - 70*x^30 + 1289*x^27 + 2848*x^24 - 33978*x^21 + 415031*x^18 - 2654421*x^15 + 8225230*x^12 + 50229628*x^9 + 90247619*x^6 + 275010364*x^3 + 594823321);
 
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{105}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, 3.3.3969.1, \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(\zeta_{9})\), 6.0.47258883.1, 6.0.47258883.2, 6.0.64827.1, 6.0.281302875.3, 6.6.843908625.1, 6.0.13783840875.2, 6.6.41351522625.2, 6.0.13783840875.1, 6.6.41351522625.1, 6.0.2100875.1, 6.6.56723625.1, 9.9.62523502209.1, 12.0.712181767349390625.1, 12.0.1709948423405886890625.5, 12.0.1709948423405886890625.6, 12.0.3217569633140625.2, 18.0.105548084868928352751387.1, 18.0.2618850774742652270958169921875.4, 18.18.70708970918051611315870587890625.1

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 ${\href{/padicField/2.6.0.1}{6} }^{6}$ R R R ${\href{/padicField/11.6.0.1}{6} }^{6}$ ${\href{/padicField/13.3.0.1}{3} }^{12}$ ${\href{/padicField/17.6.0.1}{6} }^{6}$ ${\href{/padicField/19.6.0.1}{6} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{6}$ ${\href{/padicField/29.6.0.1}{6} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{6}$ ${\href{/padicField/41.6.0.1}{6} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{6}$ ${\href{/padicField/53.6.0.1}{6} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{6}$

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
\(3\) Copy content Toggle raw display Deg $18$$6$$3$$27$
Deg $18$$6$$3$$27$
\(5\) Copy content Toggle raw display 5.12.6.1$x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
\(7\) Copy content Toggle raw display 7.18.15.1$x^{18} + 315 x^{12} + 9555 x^{6} + 76489$$6$$3$$15$$C_6 \times C_3$$[\ ]_{6}^{3}$
7.18.15.1$x^{18} + 315 x^{12} + 9555 x^{6} + 76489$$6$$3$$15$$C_6 \times C_3$$[\ ]_{6}^{3}$