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Properties

Label 12.0.597...136.1
Degree $12$
Signature $[0, 6]$
Discriminant $5.974\times 10^{33}$
Root discriminant \(652.66\)
Ramified primes $2,7,19,41$
Class number $24147396$ (GRH)
Class group [3, 3, 3, 42, 21294] (GRH)
Galois group $C_6\times S_3$ (as 12T18)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 223*x^10 + 9112*x^9 - 21670*x^8 - 26020*x^7 + 7804782*x^6 - 68604576*x^5 + 539755077*x^4 - 2463841322*x^3 + 8931195747*x^2 - 20412720936*x + 20785096368)
 
gp: K = bnfinit(y^12 - 2*y^11 + 223*y^10 + 9112*y^9 - 21670*y^8 - 26020*y^7 + 7804782*y^6 - 68604576*y^5 + 539755077*y^4 - 2463841322*y^3 + 8931195747*y^2 - 20412720936*y + 20785096368, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 + 223*x^10 + 9112*x^9 - 21670*x^8 - 26020*x^7 + 7804782*x^6 - 68604576*x^5 + 539755077*x^4 - 2463841322*x^3 + 8931195747*x^2 - 20412720936*x + 20785096368);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^11 + 223*x^10 + 9112*x^9 - 21670*x^8 - 26020*x^7 + 7804782*x^6 - 68604576*x^5 + 539755077*x^4 - 2463841322*x^3 + 8931195747*x^2 - 20412720936*x + 20785096368)
 

\( x^{12} - 2 x^{11} + 223 x^{10} + 9112 x^{9} - 21670 x^{8} - 26020 x^{7} + 7804782 x^{6} + \cdots + 20785096368 \) Copy content Toggle raw display

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

Invariants

Degree:  $12$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(5973821891382281228217152535003136\) \(\medspace = 2^{18}\cdot 7^{10}\cdot 19^{8}\cdot 41^{6}\) 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:  \(652.66\)
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:  $2^{3/2}7^{5/6}19^{2/3}41^{1/2}\approx 652.6610407652373$
Ramified primes:   \(2\), \(7\), \(19\), \(41\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $6$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{32}$

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

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a$, $\frac{1}{8}a^{5}+\frac{3}{8}a-\frac{1}{2}$, $\frac{1}{16}a^{6}-\frac{1}{16}a^{5}-\frac{5}{16}a^{2}+\frac{1}{16}a+\frac{1}{4}$, $\frac{1}{32}a^{7}-\frac{1}{32}a^{6}-\frac{5}{32}a^{3}+\frac{1}{32}a^{2}+\frac{1}{8}a$, $\frac{1}{192}a^{8}+\frac{1}{64}a^{6}+\frac{1}{48}a^{5}-\frac{5}{192}a^{4}-\frac{3}{16}a^{3}-\frac{79}{192}a^{2}+\frac{1}{3}a+\frac{1}{4}$, $\frac{1}{3072}a^{9}+\frac{7}{3072}a^{8}+\frac{9}{1024}a^{7}+\frac{73}{3072}a^{6}-\frac{97}{3072}a^{5}-\frac{359}{3072}a^{4}-\frac{547}{3072}a^{3}-\frac{499}{1024}a^{2}-\frac{163}{384}a-\frac{19}{64}$, $\frac{1}{34246656}a^{10}+\frac{169}{1070208}a^{9}+\frac{6023}{5707776}a^{8}-\frac{38261}{8561664}a^{7}-\frac{6415}{356736}a^{6}+\frac{7}{1070208}a^{5}+\frac{1353847}{17123328}a^{4}-\frac{1697}{9216}a^{3}-\frac{16772713}{34246656}a^{2}+\frac{409033}{1426944}a-\frac{59961}{237824}$, $\frac{1}{20\!\cdots\!92}a^{11}-\frac{23\!\cdots\!83}{20\!\cdots\!92}a^{10}-\frac{18\!\cdots\!03}{11\!\cdots\!44}a^{9}-\frac{28\!\cdots\!53}{10\!\cdots\!96}a^{8}+\frac{20\!\cdots\!09}{19\!\cdots\!24}a^{7}+\frac{14\!\cdots\!77}{12\!\cdots\!12}a^{6}-\frac{23\!\cdots\!57}{10\!\cdots\!96}a^{5}+\frac{84\!\cdots\!25}{10\!\cdots\!96}a^{4}+\frac{15\!\cdots\!39}{20\!\cdots\!92}a^{3}-\frac{43\!\cdots\!63}{68\!\cdots\!64}a^{2}+\frac{60\!\cdots\!39}{95\!\cdots\!12}a+\frac{12\!\cdots\!53}{47\!\cdots\!56}$ 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:  $2$, $3$

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{42}\times C_{21294}$, which has order $24147396$ (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:  $5$
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:   $\frac{78\!\cdots\!83}{34\!\cdots\!32}a^{11}+\frac{60\!\cdots\!41}{11\!\cdots\!44}a^{10}+\frac{70\!\cdots\!31}{17\!\cdots\!16}a^{9}+\frac{38\!\cdots\!77}{17\!\cdots\!16}a^{8}+\frac{20\!\cdots\!61}{85\!\cdots\!08}a^{7}-\frac{24\!\cdots\!61}{21\!\cdots\!52}a^{6}+\frac{23\!\cdots\!01}{17\!\cdots\!16}a^{5}-\frac{94\!\cdots\!73}{17\!\cdots\!16}a^{4}+\frac{38\!\cdots\!69}{34\!\cdots\!32}a^{3}+\frac{13\!\cdots\!73}{34\!\cdots\!32}a^{2}-\frac{31\!\cdots\!11}{14\!\cdots\!68}a+\frac{71\!\cdots\!33}{23\!\cdots\!28}$, $\frac{68\!\cdots\!07}{88\!\cdots\!12}a^{11}+\frac{53\!\cdots\!97}{88\!\cdots\!12}a^{10}+\frac{26\!\cdots\!89}{14\!\cdots\!52}a^{9}+\frac{37\!\cdots\!43}{44\!\cdots\!56}a^{8}+\frac{41\!\cdots\!01}{74\!\cdots\!76}a^{7}+\frac{18\!\cdots\!23}{13\!\cdots\!08}a^{6}+\frac{16\!\cdots\!41}{44\!\cdots\!56}a^{5}-\frac{48\!\cdots\!31}{44\!\cdots\!56}a^{4}+\frac{46\!\cdots\!73}{88\!\cdots\!12}a^{3}-\frac{17\!\cdots\!27}{32\!\cdots\!56}a^{2}-\frac{49\!\cdots\!63}{12\!\cdots\!96}a+\frac{27\!\cdots\!97}{20\!\cdots\!16}$, $\frac{15\!\cdots\!77}{88\!\cdots\!12}a^{11}-\frac{45\!\cdots\!89}{44\!\cdots\!56}a^{10}+\frac{17\!\cdots\!21}{49\!\cdots\!84}a^{9}+\frac{40\!\cdots\!45}{27\!\cdots\!16}a^{8}-\frac{68\!\cdots\!07}{61\!\cdots\!48}a^{7}-\frac{41\!\cdots\!45}{11\!\cdots\!64}a^{6}+\frac{75\!\cdots\!35}{44\!\cdots\!56}a^{5}-\frac{17\!\cdots\!57}{11\!\cdots\!64}a^{4}+\frac{77\!\cdots\!23}{88\!\cdots\!12}a^{3}-\frac{45\!\cdots\!65}{14\!\cdots\!52}a^{2}+\frac{46\!\cdots\!51}{77\!\cdots\!56}a-\frac{51\!\cdots\!45}{10\!\cdots\!08}$, $\frac{18\!\cdots\!83}{98\!\cdots\!68}a^{11}-\frac{91\!\cdots\!43}{29\!\cdots\!04}a^{10}+\frac{12\!\cdots\!45}{14\!\cdots\!52}a^{9}+\frac{95\!\cdots\!75}{16\!\cdots\!28}a^{8}-\frac{10\!\cdots\!13}{74\!\cdots\!76}a^{7}+\frac{21\!\cdots\!63}{10\!\cdots\!08}a^{6}-\frac{23\!\cdots\!17}{14\!\cdots\!52}a^{5}+\frac{15\!\cdots\!53}{14\!\cdots\!52}a^{4}-\frac{13\!\cdots\!57}{29\!\cdots\!04}a^{3}+\frac{44\!\cdots\!67}{29\!\cdots\!04}a^{2}-\frac{13\!\cdots\!15}{41\!\cdots\!32}a+\frac{63\!\cdots\!67}{20\!\cdots\!16}$, $\frac{27\!\cdots\!45}{11\!\cdots\!64}a^{11}+\frac{12\!\cdots\!31}{88\!\cdots\!12}a^{10}-\frac{64\!\cdots\!73}{12\!\cdots\!96}a^{9}-\frac{92\!\cdots\!73}{44\!\cdots\!56}a^{8}+\frac{35\!\cdots\!87}{24\!\cdots\!92}a^{7}+\frac{43\!\cdots\!09}{11\!\cdots\!64}a^{6}-\frac{21\!\cdots\!31}{11\!\cdots\!64}a^{5}+\frac{10\!\cdots\!85}{44\!\cdots\!56}a^{4}-\frac{34\!\cdots\!63}{22\!\cdots\!28}a^{3}+\frac{25\!\cdots\!99}{29\!\cdots\!04}a^{2}-\frac{29\!\cdots\!63}{12\!\cdots\!96}a+\frac{14\!\cdots\!95}{20\!\cdots\!16}$ 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:  \( 20248812.900721215 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 20248812.900721215 \cdot 24147396}{2\cdot\sqrt{5973821891382281228217152535003136}}\cr\approx \mathstrut & 194.622457128362 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 223*x^10 + 9112*x^9 - 21670*x^8 - 26020*x^7 + 7804782*x^6 - 68604576*x^5 + 539755077*x^4 - 2463841322*x^3 + 8931195747*x^2 - 20412720936*x + 20785096368)
 
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^12 - 2*x^11 + 223*x^10 + 9112*x^9 - 21670*x^8 - 26020*x^7 + 7804782*x^6 - 68604576*x^5 + 539755077*x^4 - 2463841322*x^3 + 8931195747*x^2 - 20412720936*x + 20785096368, 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^12 - 2*x^11 + 223*x^10 + 9112*x^9 - 21670*x^8 - 26020*x^7 + 7804782*x^6 - 68604576*x^5 + 539755077*x^4 - 2463841322*x^3 + 8931195747*x^2 - 20412720936*x + 20785096368);
 
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^12 - 2*x^11 + 223*x^10 + 9112*x^9 - 21670*x^8 - 26020*x^7 + 7804782*x^6 - 68604576*x^5 + 539755077*x^4 - 2463841322*x^3 + 8931195747*x^2 - 20412720936*x + 20785096368);
 
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

$C_6\times S_3$ (as 12T18):

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)
 
A solvable group of order 36
The 18 conjugacy class representatives for $C_6\times S_3$
Character table for $C_6\times S_3$

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{574}) \), \(\Q(\sqrt{-287}) \), \(\Q(\sqrt{-2}, \sqrt{-287})\), 6.6.77290503241874944.1

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]
 

Sibling fields

Galois closure: data not computed
Degree 18 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{6}$ ${\href{/padicField/5.6.0.1}{6} }^{2}$ R ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{6}$ R ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ ${\href{/padicField/37.6.0.1}{6} }^{2}$ R ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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 2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
\(7\) Copy content Toggle raw display 7.12.10.1$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
\(19\) Copy content Toggle raw display 19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
\(41\) Copy content Toggle raw display 41.6.3.2$x^{6} + 1681 x^{2} - 2412235$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
41.6.3.2$x^{6} + 1681 x^{2} - 2412235$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$