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Properties

Label 36.0.799...784.2
Degree $36$
Signature $[0, 18]$
Discriminant $7.996\times 10^{62}$
Root discriminant \(55.89\)
Ramified primes $2,3,19$
Class number not computed
Class group not computed
Galois group $C_2\times C_{18}$ (as 36T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 19*x^34 + 209*x^32 + 1558*x^30 + 8740*x^28 + 38095*x^26 + 132810*x^24 + 372723*x^22 + 848787*x^20 + 1558038*x^18 + 2298126*x^16 + 2670317*x^14 + 2415451*x^12 + 1629193*x^10 + 806113*x^8 + 262086*x^6 + 57399*x^4 + 5415*x^2 + 361)
 
gp: K = bnfinit(y^36 + 19*y^34 + 209*y^32 + 1558*y^30 + 8740*y^28 + 38095*y^26 + 132810*y^24 + 372723*y^22 + 848787*y^20 + 1558038*y^18 + 2298126*y^16 + 2670317*y^14 + 2415451*y^12 + 1629193*y^10 + 806113*y^8 + 262086*y^6 + 57399*y^4 + 5415*y^2 + 361, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 + 19*x^34 + 209*x^32 + 1558*x^30 + 8740*x^28 + 38095*x^26 + 132810*x^24 + 372723*x^22 + 848787*x^20 + 1558038*x^18 + 2298126*x^16 + 2670317*x^14 + 2415451*x^12 + 1629193*x^10 + 806113*x^8 + 262086*x^6 + 57399*x^4 + 5415*x^2 + 361);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 + 19*x^34 + 209*x^32 + 1558*x^30 + 8740*x^28 + 38095*x^26 + 132810*x^24 + 372723*x^22 + 848787*x^20 + 1558038*x^18 + 2298126*x^16 + 2670317*x^14 + 2415451*x^12 + 1629193*x^10 + 806113*x^8 + 262086*x^6 + 57399*x^4 + 5415*x^2 + 361)
 

\( x^{36} + 19 x^{34} + 209 x^{32} + 1558 x^{30} + 8740 x^{28} + 38095 x^{26} + 132810 x^{24} + 372723 x^{22} + \cdots + 361 \) 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:   \(799622233646074762983150698451178476894456963777140963130998784\) \(\medspace = 2^{36}\cdot 3^{18}\cdot 19^{34}\) 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:  \(55.89\)
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:  $2\cdot 3^{1/2}19^{17/18}\approx 55.88591389129187$
Ramified primes:   \(2\), \(3\), \(19\) 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:  \(228=2^{2}\cdot 3\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{228}(1,·)$, $\chi_{228}(5,·)$, $\chi_{228}(137,·)$, $\chi_{228}(143,·)$, $\chi_{228}(17,·)$, $\chi_{228}(149,·)$, $\chi_{228}(151,·)$, $\chi_{228}(25,·)$, $\chi_{228}(155,·)$, $\chi_{228}(157,·)$, $\chi_{228}(31,·)$, $\chi_{228}(161,·)$, $\chi_{228}(167,·)$, $\chi_{228}(169,·)$, $\chi_{228}(49,·)$, $\chi_{228}(179,·)$, $\chi_{228}(59,·)$, $\chi_{228}(61,·)$, $\chi_{228}(67,·)$, $\chi_{228}(197,·)$, $\chi_{228}(71,·)$, $\chi_{228}(73,·)$, $\chi_{228}(203,·)$, $\chi_{228}(77,·)$, $\chi_{228}(79,·)$, $\chi_{228}(211,·)$, $\chi_{228}(85,·)$, $\chi_{228}(91,·)$, $\chi_{228}(223,·)$, $\chi_{228}(227,·)$, $\chi_{228}(101,·)$, $\chi_{228}(103,·)$, $\chi_{228}(107,·)$, $\chi_{228}(121,·)$, $\chi_{228}(125,·)$, $\chi_{228}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{19}a^{18}$, $\frac{1}{19}a^{19}$, $\frac{1}{19}a^{20}$, $\frac{1}{19}a^{21}$, $\frac{1}{19}a^{22}$, $\frac{1}{19}a^{23}$, $\frac{1}{19}a^{24}$, $\frac{1}{19}a^{25}$, $\frac{1}{703}a^{26}+\frac{3}{703}a^{24}+\frac{12}{703}a^{22}-\frac{1}{703}a^{18}-\frac{6}{37}a^{16}-\frac{7}{37}a^{14}-\frac{2}{37}a^{12}+\frac{10}{37}a^{10}-\frac{1}{37}a^{8}-\frac{10}{37}a^{6}+\frac{4}{37}a^{4}-\frac{3}{37}a^{2}+\frac{3}{37}$, $\frac{1}{703}a^{27}+\frac{3}{703}a^{25}+\frac{12}{703}a^{23}-\frac{1}{703}a^{19}-\frac{6}{37}a^{17}-\frac{7}{37}a^{15}-\frac{2}{37}a^{13}+\frac{10}{37}a^{11}-\frac{1}{37}a^{9}-\frac{10}{37}a^{7}+\frac{4}{37}a^{5}-\frac{3}{37}a^{3}+\frac{3}{37}a$, $\frac{1}{703}a^{28}+\frac{3}{703}a^{24}+\frac{1}{703}a^{22}-\frac{1}{703}a^{20}+\frac{11}{37}a^{16}-\frac{18}{37}a^{14}+\frac{16}{37}a^{12}+\frac{6}{37}a^{10}-\frac{7}{37}a^{8}-\frac{3}{37}a^{6}-\frac{15}{37}a^{4}+\frac{12}{37}a^{2}-\frac{9}{37}$, $\frac{1}{703}a^{29}+\frac{3}{703}a^{25}+\frac{1}{703}a^{23}-\frac{1}{703}a^{21}+\frac{11}{37}a^{17}-\frac{18}{37}a^{15}+\frac{16}{37}a^{13}+\frac{6}{37}a^{11}-\frac{7}{37}a^{9}-\frac{3}{37}a^{7}-\frac{15}{37}a^{5}+\frac{12}{37}a^{3}-\frac{9}{37}a$, $\frac{1}{703}a^{30}-\frac{8}{703}a^{24}-\frac{10}{703}a^{18}+\frac{12}{37}a^{12}+\frac{15}{37}a^{6}-\frac{9}{37}$, $\frac{1}{703}a^{31}-\frac{8}{703}a^{25}-\frac{10}{703}a^{19}+\frac{12}{37}a^{13}+\frac{15}{37}a^{7}-\frac{9}{37}a$, $\frac{1}{703}a^{32}-\frac{13}{703}a^{24}-\frac{15}{703}a^{22}-\frac{10}{703}a^{20}-\frac{8}{703}a^{18}-\frac{11}{37}a^{16}-\frac{7}{37}a^{14}-\frac{16}{37}a^{12}+\frac{6}{37}a^{10}+\frac{7}{37}a^{8}-\frac{6}{37}a^{6}-\frac{5}{37}a^{4}+\frac{4}{37}a^{2}-\frac{13}{37}$, $\frac{1}{703}a^{33}-\frac{13}{703}a^{25}-\frac{15}{703}a^{23}-\frac{10}{703}a^{21}-\frac{8}{703}a^{19}-\frac{11}{37}a^{17}-\frac{7}{37}a^{15}-\frac{16}{37}a^{13}+\frac{6}{37}a^{11}+\frac{7}{37}a^{9}-\frac{6}{37}a^{7}-\frac{5}{37}a^{5}+\frac{4}{37}a^{3}-\frac{13}{37}a$, $\frac{1}{66\!\cdots\!19}a^{34}+\frac{95\!\cdots\!18}{18\!\cdots\!79}a^{32}+\frac{18\!\cdots\!66}{34\!\cdots\!01}a^{30}+\frac{17\!\cdots\!52}{34\!\cdots\!01}a^{28}+\frac{44\!\cdots\!13}{34\!\cdots\!01}a^{26}-\frac{53\!\cdots\!64}{34\!\cdots\!01}a^{24}+\frac{26\!\cdots\!23}{34\!\cdots\!01}a^{22}+\frac{32\!\cdots\!72}{34\!\cdots\!01}a^{20}+\frac{91\!\cdots\!07}{34\!\cdots\!01}a^{18}-\frac{13\!\cdots\!42}{34\!\cdots\!01}a^{16}-\frac{30\!\cdots\!48}{18\!\cdots\!79}a^{14}-\frac{43\!\cdots\!58}{18\!\cdots\!79}a^{12}-\frac{71\!\cdots\!88}{18\!\cdots\!79}a^{10}-\frac{55\!\cdots\!47}{18\!\cdots\!79}a^{8}+\frac{26\!\cdots\!46}{18\!\cdots\!79}a^{6}-\frac{34\!\cdots\!66}{18\!\cdots\!79}a^{4}-\frac{38\!\cdots\!15}{18\!\cdots\!79}a^{2}+\frac{65\!\cdots\!31}{18\!\cdots\!79}$, $\frac{1}{66\!\cdots\!19}a^{35}+\frac{95\!\cdots\!18}{18\!\cdots\!79}a^{33}+\frac{18\!\cdots\!66}{34\!\cdots\!01}a^{31}+\frac{17\!\cdots\!52}{34\!\cdots\!01}a^{29}+\frac{44\!\cdots\!13}{34\!\cdots\!01}a^{27}-\frac{53\!\cdots\!64}{34\!\cdots\!01}a^{25}+\frac{26\!\cdots\!23}{34\!\cdots\!01}a^{23}+\frac{32\!\cdots\!72}{34\!\cdots\!01}a^{21}+\frac{91\!\cdots\!07}{34\!\cdots\!01}a^{19}-\frac{13\!\cdots\!42}{34\!\cdots\!01}a^{17}-\frac{30\!\cdots\!48}{18\!\cdots\!79}a^{15}-\frac{43\!\cdots\!58}{18\!\cdots\!79}a^{13}-\frac{71\!\cdots\!88}{18\!\cdots\!79}a^{11}-\frac{55\!\cdots\!47}{18\!\cdots\!79}a^{9}+\frac{26\!\cdots\!46}{18\!\cdots\!79}a^{7}-\frac{34\!\cdots\!66}{18\!\cdots\!79}a^{5}-\frac{38\!\cdots\!15}{18\!\cdots\!79}a^{3}+\frac{65\!\cdots\!31}{18\!\cdots\!79}a$ 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:  $17$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{26829997518280183021}{66495926797178861491219} a^{34} + \frac{26598244133079038772}{3499785620904150604801} a^{32} + \frac{290816878133015963118}{3499785620904150604801} a^{30} + \frac{2153288311705603483287}{3499785620904150604801} a^{28} + \frac{11998687105984961447667}{3499785620904150604801} a^{26} + \frac{51899600479398829656543}{3499785620904150604801} a^{24} + \frac{4849522942561243997067}{94588800564977043373} a^{22} + \frac{498605177959280803164861}{3499785620904150604801} a^{20} + \frac{1122593033140817945162022}{3499785620904150604801} a^{18} + \frac{2031702862437939905122865}{3499785620904150604801} a^{16} + \frac{155072516601073134700314}{184199243205481610779} a^{14} + \frac{176194500030210489789879}{184199243205481610779} a^{12} + \frac{155039966107495776853800}{184199243205481610779} a^{10} + \frac{100425776080331302112142}{184199243205481610779} a^{8} + \frac{47425533042550358380716}{184199243205481610779} a^{6} + \frac{14082041180781203080581}{184199243205481610779} a^{4} + \frac{3119154647880984572367}{184199243205481610779} a^{2} + \frac{291248148377115816489}{184199243205481610779} \)  (order $6$) 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 + 19*x^34 + 209*x^32 + 1558*x^30 + 8740*x^28 + 38095*x^26 + 132810*x^24 + 372723*x^22 + 848787*x^20 + 1558038*x^18 + 2298126*x^16 + 2670317*x^14 + 2415451*x^12 + 1629193*x^10 + 806113*x^8 + 262086*x^6 + 57399*x^4 + 5415*x^2 + 361)
 
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 + 19*x^34 + 209*x^32 + 1558*x^30 + 8740*x^28 + 38095*x^26 + 132810*x^24 + 372723*x^22 + 848787*x^20 + 1558038*x^18 + 2298126*x^16 + 2670317*x^14 + 2415451*x^12 + 1629193*x^10 + 806113*x^8 + 262086*x^6 + 57399*x^4 + 5415*x^2 + 361, 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 + 19*x^34 + 209*x^32 + 1558*x^30 + 8740*x^28 + 38095*x^26 + 132810*x^24 + 372723*x^22 + 848787*x^20 + 1558038*x^18 + 2298126*x^16 + 2670317*x^14 + 2415451*x^12 + 1629193*x^10 + 806113*x^8 + 262086*x^6 + 57399*x^4 + 5415*x^2 + 361);
 
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 + 19*x^34 + 209*x^32 + 1558*x^30 + 8740*x^28 + 38095*x^26 + 132810*x^24 + 372723*x^22 + 848787*x^20 + 1558038*x^18 + 2298126*x^16 + 2670317*x^14 + 2415451*x^12 + 1629193*x^10 + 806113*x^8 + 262086*x^6 + 57399*x^4 + 5415*x^2 + 361);
 
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

$C_2\times C_{18}$ (as 36T2):

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 $C_2\times C_{18}$
Character table for $C_2\times C_{18}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{19}) \), \(\Q(\sqrt{-57}) \), 3.3.361.1, \(\Q(\sqrt{-3}, \sqrt{19})\), 6.0.3518667.1, 6.6.158470336.1, 6.0.4278699072.1, \(\Q(\zeta_{19})^+\), 12.0.18307265748733661184.2, 18.0.5677392343251487443465123.1, \(\Q(\zeta_{76})^+\), 18.0.28277592430157040563214702870528.1

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 $18^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{6}$ ${\href{/padicField/11.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$ R $18^{2}$ $18^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{12}$ ${\href{/padicField/37.2.0.1}{2} }^{18}$ $18^{2}$ $18^{2}$ $18^{2}$ $18^{2}$ $18^{2}$

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 $36$$2$$18$$36$
\(3\) Copy content Toggle raw display Deg $18$$2$$9$$9$
Deg $18$$2$$9$$9$
\(19\) Copy content Toggle raw display 19.18.17.1$x^{18} + 342$$18$$1$$17$$C_{18}$$[\ ]_{18}$
19.18.17.1$x^{18} + 342$$18$$1$$17$$C_{18}$$[\ ]_{18}$