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Properties

Label 12.0.120...952.2
Degree $12$
Signature $[0, 6]$
Discriminant $1.205\times 10^{29}$
Root discriminant \(265.11\)
Ramified primes $2,3,7,17$
Class number $105996$ (GRH)
Class group [22, 4818] (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 33*x^10 - 140*x^9 + 1575*x^8 + 8820*x^7 - 709*x^6 - 206640*x^5 - 346269*x^4 + 3026660*x^3 + 21758073*x^2 + 55541430*x + 71323433)
 
gp: K = bnfinit(y^12 - 33*y^10 - 140*y^9 + 1575*y^8 + 8820*y^7 - 709*y^6 - 206640*y^5 - 346269*y^4 + 3026660*y^3 + 21758073*y^2 + 55541430*y + 71323433, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 33*x^10 - 140*x^9 + 1575*x^8 + 8820*x^7 - 709*x^6 - 206640*x^5 - 346269*x^4 + 3026660*x^3 + 21758073*x^2 + 55541430*x + 71323433);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 33*x^10 - 140*x^9 + 1575*x^8 + 8820*x^7 - 709*x^6 - 206640*x^5 - 346269*x^4 + 3026660*x^3 + 21758073*x^2 + 55541430*x + 71323433)
 

\( x^{12} - 33 x^{10} - 140 x^{9} + 1575 x^{8} + 8820 x^{7} - 709 x^{6} - 206640 x^{5} - 346269 x^{4} + \cdots + 71323433 \) 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:   \(120538181723674419070377725952\) \(\medspace = 2^{12}\cdot 3^{16}\cdot 7^{8}\cdot 17^{9}\) 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:  \(265.11\)
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:  $2\cdot 3^{4/3}7^{2/3}17^{3/4}\approx 265.1105758191317$
Ramified primes:   \(2\), \(3\), \(7\), \(17\) 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:  \(\Q(\sqrt{17}) \)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4284=2^{2}\cdot 3^{2}\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{4284}(1,·)$, $\chi_{4284}(1891,·)$, $\chi_{4284}(2209,·)$, $\chi_{4284}(1135,·)$, $\chi_{4284}(3469,·)$, $\chi_{4284}(2095,·)$, $\chi_{4284}(3025,·)$, $\chi_{4284}(205,·)$, $\chi_{4284}(1075,·)$, $\chi_{4284}(1339,·)$, $\chi_{4284}(3229,·)$, $\chi_{4284}(319,·)$$\rbrace$
This is a CM field.
Reflex fields:  4.0.78608.1$^{2}$, 12.0.120538181723674419070377725952.2$^{30}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{3799929922}a^{10}-\frac{11003305}{1899964961}a^{9}+\frac{436200953}{1899964961}a^{8}-\frac{799634671}{3799929922}a^{7}+\frac{1533878101}{3799929922}a^{6}-\frac{1647538615}{3799929922}a^{5}+\frac{939853093}{3799929922}a^{4}-\frac{947808605}{1899964961}a^{3}+\frac{418642041}{3799929922}a^{2}+\frac{172426875}{3799929922}a+\frac{383965168}{1899964961}$, $\frac{1}{37\!\cdots\!18}a^{11}-\frac{6111613513}{18\!\cdots\!09}a^{10}+\frac{39\!\cdots\!02}{18\!\cdots\!09}a^{9}-\frac{18\!\cdots\!41}{37\!\cdots\!18}a^{8}-\frac{10\!\cdots\!93}{37\!\cdots\!18}a^{7}+\frac{17\!\cdots\!33}{37\!\cdots\!18}a^{6}-\frac{11\!\cdots\!29}{37\!\cdots\!18}a^{5}+\frac{62\!\cdots\!46}{18\!\cdots\!09}a^{4}-\frac{16\!\cdots\!53}{37\!\cdots\!18}a^{3}+\frac{56\!\cdots\!13}{37\!\cdots\!18}a^{2}+\frac{89\!\cdots\!63}{18\!\cdots\!09}a-\frac{73\!\cdots\!28}{18\!\cdots\!09}$ 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

$C_{22}\times C_{4818}$, which has order $105996$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Relative class number: $35332$ (assuming GRH)

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:   $\frac{266472244369404}{18\!\cdots\!09}a^{11}-\frac{22\!\cdots\!74}{18\!\cdots\!09}a^{10}-\frac{105209114730024}{18\!\cdots\!09}a^{9}+\frac{24\!\cdots\!51}{18\!\cdots\!09}a^{8}+\frac{39\!\cdots\!96}{18\!\cdots\!09}a^{7}-\frac{15\!\cdots\!88}{18\!\cdots\!09}a^{6}-\frac{44\!\cdots\!18}{18\!\cdots\!09}a^{5}-\frac{52\!\cdots\!16}{18\!\cdots\!09}a^{4}+\frac{17\!\cdots\!98}{18\!\cdots\!09}a^{3}+\frac{17\!\cdots\!93}{18\!\cdots\!09}a^{2}-\frac{21\!\cdots\!64}{18\!\cdots\!09}a-\frac{36\!\cdots\!08}{18\!\cdots\!09}$, $\frac{231807887141616}{18\!\cdots\!09}a^{11}-\frac{20\!\cdots\!44}{18\!\cdots\!09}a^{10}-\frac{47336365467116}{18\!\cdots\!09}a^{9}+\frac{22\!\cdots\!61}{18\!\cdots\!09}a^{8}+\frac{34\!\cdots\!24}{18\!\cdots\!09}a^{7}-\frac{13\!\cdots\!38}{18\!\cdots\!09}a^{6}-\frac{40\!\cdots\!52}{18\!\cdots\!09}a^{5}-\frac{46\!\cdots\!21}{18\!\cdots\!09}a^{4}+\frac{15\!\cdots\!72}{18\!\cdots\!09}a^{3}+\frac{16\!\cdots\!38}{18\!\cdots\!09}a^{2}-\frac{16\!\cdots\!36}{18\!\cdots\!09}a-\frac{43\!\cdots\!44}{18\!\cdots\!09}$, $\frac{7560}{1899964961}a^{11}-\frac{14064}{1899964961}a^{10}-\frac{297360}{1899964961}a^{9}-\frac{113250}{1899964961}a^{8}+\frac{13786920}{1899964961}a^{7}+\frac{36981988}{1899964961}a^{6}-\frac{200418540}{1899964961}a^{5}-\frac{1168255170}{1899964961}a^{4}+\frac{1527535940}{1899964961}a^{3}+\frac{26201869086}{1899964961}a^{2}+\frac{82385572080}{1899964961}a+\frac{102912083971}{1899964961}$, $\frac{969746255375256}{18\!\cdots\!09}a^{11}-\frac{33\!\cdots\!60}{18\!\cdots\!09}a^{10}-\frac{29\!\cdots\!56}{18\!\cdots\!09}a^{9}+\frac{11\!\cdots\!11}{18\!\cdots\!09}a^{8}+\frac{16\!\cdots\!04}{18\!\cdots\!09}a^{7}+\frac{22\!\cdots\!34}{18\!\cdots\!09}a^{6}-\frac{23\!\cdots\!12}{18\!\cdots\!09}a^{5}-\frac{11\!\cdots\!51}{18\!\cdots\!09}a^{4}+\frac{30\!\cdots\!32}{18\!\cdots\!09}a^{3}+\frac{27\!\cdots\!72}{18\!\cdots\!09}a^{2}+\frac{78\!\cdots\!84}{18\!\cdots\!09}a+\frac{54\!\cdots\!46}{18\!\cdots\!09}$, $\frac{772602725461428}{18\!\cdots\!09}a^{11}-\frac{16\!\cdots\!46}{18\!\cdots\!09}a^{10}-\frac{29\!\cdots\!48}{18\!\cdots\!09}a^{9}-\frac{86\!\cdots\!60}{18\!\cdots\!09}a^{8}+\frac{13\!\cdots\!52}{18\!\cdots\!09}a^{7}+\frac{34\!\cdots\!22}{18\!\cdots\!09}a^{6}-\frac{19\!\cdots\!26}{18\!\cdots\!09}a^{5}-\frac{11\!\cdots\!25}{18\!\cdots\!09}a^{4}+\frac{16\!\cdots\!86}{18\!\cdots\!09}a^{3}+\frac{25\!\cdots\!89}{18\!\cdots\!09}a^{2}+\frac{79\!\cdots\!92}{18\!\cdots\!09}a+\frac{94\!\cdots\!72}{18\!\cdots\!09}$ 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:  \( 23314.354920427802 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 23314.354920427802 \cdot 105996}{2\cdot\sqrt{120538181723674419070377725952}}\cr\approx \mathstrut & 0.218977642833237 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 33*x^10 - 140*x^9 + 1575*x^8 + 8820*x^7 - 709*x^6 - 206640*x^5 - 346269*x^4 + 3026660*x^3 + 21758073*x^2 + 55541430*x + 71323433)
 
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 - 33*x^10 - 140*x^9 + 1575*x^8 + 8820*x^7 - 709*x^6 - 206640*x^5 - 346269*x^4 + 3026660*x^3 + 21758073*x^2 + 55541430*x + 71323433, 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 - 33*x^10 - 140*x^9 + 1575*x^8 + 8820*x^7 - 709*x^6 - 206640*x^5 - 346269*x^4 + 3026660*x^3 + 21758073*x^2 + 55541430*x + 71323433);
 
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 - 33*x^10 - 140*x^9 + 1575*x^8 + 8820*x^7 - 709*x^6 - 206640*x^5 - 346269*x^4 + 3026660*x^3 + 21758073*x^2 + 55541430*x + 71323433);
 
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

\(\Q(\sqrt{17}) \), 3.3.3969.2, 4.0.78608.1, 6.6.77394297393.2

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 ${\href{/padicField/5.4.0.1}{4} }^{3}$ R ${\href{/padicField/11.4.0.1}{4} }^{3}$ ${\href{/padicField/13.3.0.1}{3} }^{4}$ R ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{3}$ ${\href{/padicField/29.12.0.1}{12} }$ ${\href{/padicField/31.12.0.1}{12} }$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.3.0.1}{3} }^{4}$

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
\(2\) Copy content Toggle raw display 2.6.6.5$x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$$2$$3$$6$$C_6$$[2]^{3}$
\(3\) Copy content Toggle raw display 3.12.16.25$x^{12} + 24 x^{11} + 216 x^{10} + 804 x^{9} + 216 x^{8} - 6480 x^{7} - 11610 x^{6} + 16200 x^{5} + 48600 x^{4} + 33156 x^{3} + 198936 x^{2} + 190593$$3$$4$$16$$C_{12}$$[2]^{4}$
\(7\) Copy content Toggle raw display 7.12.8.2$x^{12} - 70 x^{9} + 1519 x^{6} - 4802 x^{3} + 21609$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
\(17\) Copy content Toggle raw display 17.12.9.1$x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$