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Properties

Label 12.0.112...125.1
Degree $12$
Signature $[0, 6]$
Discriminant $1.120\times 10^{23}$
Root discriminant \(83.32\)
Ramified primes $3,5,23$
Class number $17540$ (GRH)
Class group [2, 8770] (GRH)
Galois group $C_{12}$ (as 12T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 168*x^10 - x^9 + 9774*x^8 - 1701*x^7 + 257648*x^6 - 137187*x^5 + 3288795*x^4 - 3206849*x^3 + 21932874*x^2 - 17829192*x + 76909321)
 
gp: K = bnfinit(y^12 + 168*y^10 - y^9 + 9774*y^8 - 1701*y^7 + 257648*y^6 - 137187*y^5 + 3288795*y^4 - 3206849*y^3 + 21932874*y^2 - 17829192*y + 76909321, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 168*x^10 - x^9 + 9774*x^8 - 1701*x^7 + 257648*x^6 - 137187*x^5 + 3288795*x^4 - 3206849*x^3 + 21932874*x^2 - 17829192*x + 76909321);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 + 168*x^10 - x^9 + 9774*x^8 - 1701*x^7 + 257648*x^6 - 137187*x^5 + 3288795*x^4 - 3206849*x^3 + 21932874*x^2 - 17829192*x + 76909321)
 

\( x^{12} + 168 x^{10} - x^{9} + 9774 x^{8} - 1701 x^{7} + 257648 x^{6} - 137187 x^{5} + 3288795 x^{4} + \cdots + 76909321 \) 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:   \(112015891613143986328125\) \(\medspace = 3^{18}\cdot 5^{9}\cdot 23^{6}\) 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:  \(83.32\)
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:  $3^{3/2}5^{3/4}23^{1/2}\approx 83.32461263118387$
Ramified primes:   \(3\), \(5\), \(23\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1035=3^{2}\cdot 5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{1035}(1,·)$, $\chi_{1035}(482,·)$, $\chi_{1035}(68,·)$, $\chi_{1035}(137,·)$, $\chi_{1035}(139,·)$, $\chi_{1035}(413,·)$, $\chi_{1035}(691,·)$, $\chi_{1035}(758,·)$, $\chi_{1035}(484,·)$, $\chi_{1035}(346,·)$, $\chi_{1035}(827,·)$, $\chi_{1035}(829,·)$$\rbrace$
This is a CM field.
Reflex fields:  4.0.595125.1$^{2}$, 12.0.112015891613143986328125.1$^{30}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{19}a^{8}+\frac{2}{19}a^{7}-\frac{8}{19}a^{6}+\frac{9}{19}a^{5}+\frac{1}{19}a^{4}+\frac{6}{19}a^{3}+\frac{6}{19}a^{2}+\frac{2}{19}a$, $\frac{1}{209}a^{9}-\frac{69}{209}a^{7}-\frac{51}{209}a^{6}+\frac{2}{209}a^{5}+\frac{42}{209}a^{4}-\frac{63}{209}a^{3}+\frac{104}{209}a^{2}+\frac{53}{209}a+\frac{4}{11}$, $\frac{1}{209}a^{10}-\frac{3}{209}a^{8}+\frac{81}{209}a^{7}+\frac{101}{209}a^{6}+\frac{9}{209}a^{5}+\frac{3}{209}a^{4}+\frac{82}{209}a^{3}+\frac{31}{209}a^{2}-\frac{1}{209}a$, $\frac{1}{22\!\cdots\!49}a^{11}+\frac{45\!\cdots\!13}{22\!\cdots\!49}a^{10}+\frac{46\!\cdots\!20}{22\!\cdots\!49}a^{9}+\frac{17\!\cdots\!96}{22\!\cdots\!49}a^{8}-\frac{74\!\cdots\!30}{20\!\cdots\!59}a^{7}-\frac{76\!\cdots\!37}{22\!\cdots\!49}a^{6}+\frac{74\!\cdots\!44}{22\!\cdots\!49}a^{5}-\frac{61\!\cdots\!86}{22\!\cdots\!49}a^{4}-\frac{19\!\cdots\!10}{22\!\cdots\!49}a^{3}+\frac{33\!\cdots\!79}{22\!\cdots\!49}a^{2}-\frac{67\!\cdots\!16}{22\!\cdots\!49}a-\frac{45\!\cdots\!72}{12\!\cdots\!71}$ 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

$C_{2}\times C_{8770}$, which has order $17540$ (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:   $\frac{26\!\cdots\!80}{10\!\cdots\!21}a^{11}-\frac{574257336646374}{10\!\cdots\!21}a^{10}+\frac{41\!\cdots\!15}{10\!\cdots\!21}a^{9}+\frac{29\!\cdots\!66}{10\!\cdots\!21}a^{8}+\frac{20\!\cdots\!43}{10\!\cdots\!21}a^{7}+\frac{10\!\cdots\!88}{10\!\cdots\!21}a^{6}+\frac{42\!\cdots\!57}{10\!\cdots\!21}a^{5}-\frac{45\!\cdots\!70}{10\!\cdots\!21}a^{4}+\frac{33\!\cdots\!24}{10\!\cdots\!21}a^{3}-\frac{81\!\cdots\!30}{10\!\cdots\!21}a^{2}+\frac{95\!\cdots\!75}{10\!\cdots\!21}a+\frac{96\!\cdots\!71}{53\!\cdots\!59}$, $\frac{25\!\cdots\!80}{16\!\cdots\!41}a^{11}+\frac{17\!\cdots\!29}{16\!\cdots\!41}a^{10}+\frac{37\!\cdots\!10}{16\!\cdots\!41}a^{9}+\frac{27\!\cdots\!10}{16\!\cdots\!41}a^{8}+\frac{17\!\cdots\!20}{16\!\cdots\!41}a^{7}+\frac{10\!\cdots\!05}{16\!\cdots\!41}a^{6}+\frac{27\!\cdots\!59}{15\!\cdots\!31}a^{5}+\frac{93\!\cdots\!90}{16\!\cdots\!41}a^{4}+\frac{18\!\cdots\!75}{16\!\cdots\!41}a^{3}+\frac{14\!\cdots\!95}{16\!\cdots\!41}a^{2}+\frac{43\!\cdots\!15}{16\!\cdots\!41}a+\frac{14\!\cdots\!69}{87\!\cdots\!39}$, $\frac{25\!\cdots\!80}{16\!\cdots\!41}a^{11}+\frac{17\!\cdots\!29}{16\!\cdots\!41}a^{10}+\frac{37\!\cdots\!10}{16\!\cdots\!41}a^{9}+\frac{27\!\cdots\!10}{16\!\cdots\!41}a^{8}+\frac{17\!\cdots\!20}{16\!\cdots\!41}a^{7}+\frac{10\!\cdots\!05}{16\!\cdots\!41}a^{6}+\frac{27\!\cdots\!59}{15\!\cdots\!31}a^{5}+\frac{93\!\cdots\!90}{16\!\cdots\!41}a^{4}+\frac{18\!\cdots\!75}{16\!\cdots\!41}a^{3}+\frac{14\!\cdots\!95}{16\!\cdots\!41}a^{2}+\frac{43\!\cdots\!15}{16\!\cdots\!41}a+\frac{23\!\cdots\!08}{87\!\cdots\!39}$, $\frac{37\!\cdots\!40}{20\!\cdots\!59}a^{11}+\frac{37\!\cdots\!64}{22\!\cdots\!49}a^{10}+\frac{60\!\cdots\!65}{22\!\cdots\!49}a^{9}+\frac{57\!\cdots\!14}{22\!\cdots\!49}a^{8}+\frac{27\!\cdots\!27}{22\!\cdots\!49}a^{7}+\frac{21\!\cdots\!72}{22\!\cdots\!49}a^{6}+\frac{48\!\cdots\!94}{20\!\cdots\!59}a^{5}+\frac{16\!\cdots\!90}{22\!\cdots\!49}a^{4}+\frac{46\!\cdots\!31}{22\!\cdots\!49}a^{3}-\frac{14\!\cdots\!10}{22\!\cdots\!49}a^{2}+\frac{15\!\cdots\!90}{22\!\cdots\!49}a-\frac{52\!\cdots\!82}{12\!\cdots\!71}$, $\frac{81\!\cdots\!98}{22\!\cdots\!49}a^{11}+\frac{23\!\cdots\!27}{22\!\cdots\!49}a^{10}+\frac{14\!\cdots\!28}{22\!\cdots\!49}a^{9}+\frac{36\!\cdots\!88}{22\!\cdots\!49}a^{8}+\frac{94\!\cdots\!61}{22\!\cdots\!49}a^{7}+\frac{17\!\cdots\!79}{22\!\cdots\!49}a^{6}+\frac{26\!\cdots\!39}{20\!\cdots\!59}a^{5}+\frac{21\!\cdots\!20}{22\!\cdots\!49}a^{4}+\frac{38\!\cdots\!18}{22\!\cdots\!49}a^{3}-\frac{32\!\cdots\!30}{22\!\cdots\!49}a^{2}+\frac{13\!\cdots\!13}{22\!\cdots\!49}a-\frac{29\!\cdots\!09}{12\!\cdots\!71}$ 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:  \( 201.000834787 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 201.000834787 \cdot 17540}{2\cdot\sqrt{112015891613143986328125}}\cr\approx \mathstrut & 0.324068519143 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 + 168*x^10 - x^9 + 9774*x^8 - 1701*x^7 + 257648*x^6 - 137187*x^5 + 3288795*x^4 - 3206849*x^3 + 21932874*x^2 - 17829192*x + 76909321)
 
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 + 168*x^10 - x^9 + 9774*x^8 - 1701*x^7 + 257648*x^6 - 137187*x^5 + 3288795*x^4 - 3206849*x^3 + 21932874*x^2 - 17829192*x + 76909321, 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 + 168*x^10 - x^9 + 9774*x^8 - 1701*x^7 + 257648*x^6 - 137187*x^5 + 3288795*x^4 - 3206849*x^3 + 21932874*x^2 - 17829192*x + 76909321);
 
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 + 168*x^10 - x^9 + 9774*x^8 - 1701*x^7 + 257648*x^6 - 137187*x^5 + 3288795*x^4 - 3206849*x^3 + 21932874*x^2 - 17829192*x + 76909321);
 
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

$C_{12}$ (as 12T1):

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 cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 4.0.595125.1, 6.6.820125.1

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 ${\href{/padicField/2.12.0.1}{12} }$ R R ${\href{/padicField/7.12.0.1}{12} }$ ${\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.12.0.1}{12} }$ ${\href{/padicField/17.4.0.1}{4} }^{3}$ ${\href{/padicField/19.1.0.1}{1} }^{12}$ R ${\href{/padicField/29.3.0.1}{3} }^{4}$ ${\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.3.0.1}{3} }^{4}$

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 3.12.18.74$x^{12} + 12 x^{11} + 42 x^{10} + 42 x^{9} + 54 x^{8} + 18 x^{7} - 33 x^{6} - 252 x^{5} - 126 x^{3} + 882$$6$$2$$18$$C_{12}$$[2]_{2}^{2}$
\(5\) Copy content Toggle raw display 5.12.9.2$x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
\(23\) Copy content Toggle raw display 23.12.6.2$x^{12} + 529 x^{8} - 109503 x^{6} + 2518569 x^{4} - 6436343 x^{2} + 740179445$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$