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Properties

Label 12.0.253...528.1
Degree $12$
Signature $[0, 6]$
Discriminant $2.536\times 10^{25}$
Root discriminant \(130.92\)
Ramified primes $2,13,17$
Class number $8788$ (GRH)
Class group [26, 338] (GRH)
Galois group $C_{12}$ (as 12T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 92*x^10 - 300*x^9 + 3971*x^8 - 9660*x^7 + 97334*x^6 - 167928*x^5 + 1444284*x^4 - 1443204*x^3 + 12846660*x^2 - 4211292*x + 52322057)
 
gp: K = bnfinit(y^12 - 4*y^11 + 92*y^10 - 300*y^9 + 3971*y^8 - 9660*y^7 + 97334*y^6 - 167928*y^5 + 1444284*y^4 - 1443204*y^3 + 12846660*y^2 - 4211292*y + 52322057, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 92*x^10 - 300*x^9 + 3971*x^8 - 9660*x^7 + 97334*x^6 - 167928*x^5 + 1444284*x^4 - 1443204*x^3 + 12846660*x^2 - 4211292*x + 52322057);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 92*x^10 - 300*x^9 + 3971*x^8 - 9660*x^7 + 97334*x^6 - 167928*x^5 + 1444284*x^4 - 1443204*x^3 + 12846660*x^2 - 4211292*x + 52322057)
 

\( x^{12} - 4 x^{11} + 92 x^{10} - 300 x^{9} + 3971 x^{8} - 9660 x^{7} + 97334 x^{6} - 167928 x^{5} + \cdots + 52322057 \) 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:   \(25358702738605805230358528\) \(\medspace = 2^{18}\cdot 13^{8}\cdot 17^{9}\) 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:  \(130.92\)
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:  $2^{3/2}13^{2/3}17^{3/4}\approx 130.92138351184528$
Ramified primes:   \(2\), \(13\), \(17\) 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:  \(1768=2^{3}\cdot 13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{1768}(1,·)$, $\chi_{1768}(523,·)$, $\chi_{1768}(1121,·)$, $\chi_{1768}(1225,·)$, $\chi_{1768}(939,·)$, $\chi_{1768}(1361,·)$, $\chi_{1768}(1075,·)$, $\chi_{1768}(659,·)$, $\chi_{1768}(1067,·)$, $\chi_{1768}(1665,·)$, $\chi_{1768}(1257,·)$, $\chi_{1768}(1483,·)$$\rbrace$
This is a CM field.
Reflex fields:  4.0.314432.2$^{2}$, 12.0.25358702738605805230358528.1$^{30}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{631748836}a^{10}+\frac{56632033}{631748836}a^{9}+\frac{91285361}{631748836}a^{8}-\frac{5575786}{157937209}a^{7}+\frac{63320067}{315874418}a^{6}+\frac{226502585}{631748836}a^{5}-\frac{312597209}{631748836}a^{4}+\frac{41461593}{631748836}a^{3}+\frac{40966996}{157937209}a^{2}-\frac{145574579}{631748836}a-\frac{9865938}{157937209}$, $\frac{1}{10\!\cdots\!52}a^{11}-\frac{20143395833}{54\!\cdots\!26}a^{10}-\frac{36\!\cdots\!15}{10\!\cdots\!52}a^{9}+\frac{47\!\cdots\!93}{54\!\cdots\!26}a^{8}+\frac{19\!\cdots\!25}{10\!\cdots\!52}a^{7}-\frac{15\!\cdots\!99}{10\!\cdots\!52}a^{6}+\frac{23\!\cdots\!63}{54\!\cdots\!26}a^{5}-\frac{40\!\cdots\!21}{10\!\cdots\!52}a^{4}-\frac{80\!\cdots\!37}{54\!\cdots\!26}a^{3}-\frac{62\!\cdots\!49}{54\!\cdots\!26}a^{2}+\frac{51\!\cdots\!57}{10\!\cdots\!52}a+\frac{39\!\cdots\!31}{10\!\cdots\!52}$ 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

$C_{26}\times C_{338}$, which has order $8788$ (assuming GRH)

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

Relative class number: $8788$ (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:   $\frac{7051202169360}{27\!\cdots\!13}a^{11}-\frac{9187613265100}{27\!\cdots\!13}a^{10}+\frac{347440856193576}{27\!\cdots\!13}a^{9}+\frac{319680107545237}{27\!\cdots\!13}a^{8}+\frac{64\!\cdots\!80}{27\!\cdots\!13}a^{7}+\frac{39\!\cdots\!84}{27\!\cdots\!13}a^{6}-\frac{14\!\cdots\!76}{27\!\cdots\!13}a^{5}+\frac{11\!\cdots\!18}{27\!\cdots\!13}a^{4}-\frac{20\!\cdots\!84}{27\!\cdots\!13}a^{3}+\frac{15\!\cdots\!00}{27\!\cdots\!13}a^{2}-\frac{15\!\cdots\!40}{27\!\cdots\!13}a+\frac{87\!\cdots\!42}{27\!\cdots\!13}$, $\frac{13814861071500}{27\!\cdots\!13}a^{11}+\frac{34754653454794}{27\!\cdots\!13}a^{10}+\frac{572275994956592}{27\!\cdots\!13}a^{9}+\frac{52\!\cdots\!52}{27\!\cdots\!13}a^{8}+\frac{47\!\cdots\!92}{27\!\cdots\!13}a^{7}+\frac{25\!\cdots\!54}{27\!\cdots\!13}a^{6}-\frac{21\!\cdots\!12}{27\!\cdots\!13}a^{5}+\frac{61\!\cdots\!91}{27\!\cdots\!13}a^{4}-\frac{51\!\cdots\!44}{27\!\cdots\!13}a^{3}+\frac{79\!\cdots\!24}{27\!\cdots\!13}a^{2}-\frac{27\!\cdots\!24}{27\!\cdots\!13}a+\frac{49\!\cdots\!46}{27\!\cdots\!13}$, $\frac{29450227032770}{27\!\cdots\!13}a^{11}-\frac{288486223292697}{27\!\cdots\!13}a^{10}+\frac{28\!\cdots\!21}{27\!\cdots\!13}a^{9}-\frac{37\!\cdots\!71}{54\!\cdots\!26}a^{8}+\frac{11\!\cdots\!20}{27\!\cdots\!13}a^{7}-\frac{55\!\cdots\!47}{27\!\cdots\!13}a^{6}+\frac{22\!\cdots\!04}{27\!\cdots\!13}a^{5}-\frac{17\!\cdots\!39}{54\!\cdots\!26}a^{4}+\frac{22\!\cdots\!76}{27\!\cdots\!13}a^{3}-\frac{14\!\cdots\!69}{54\!\cdots\!26}a^{2}+\frac{94\!\cdots\!30}{27\!\cdots\!13}a-\frac{73\!\cdots\!25}{54\!\cdots\!26}$, $\frac{8584163791910}{27\!\cdots\!13}a^{11}-\frac{314053263482391}{27\!\cdots\!13}a^{10}+\frac{19\!\cdots\!53}{27\!\cdots\!13}a^{9}-\frac{48\!\cdots\!49}{54\!\cdots\!26}a^{8}+\frac{10\!\cdots\!48}{27\!\cdots\!13}a^{7}-\frac{84\!\cdots\!85}{27\!\cdots\!13}a^{6}+\frac{24\!\cdots\!92}{27\!\cdots\!13}a^{5}-\frac{31\!\cdots\!57}{54\!\cdots\!26}a^{4}+\frac{29\!\cdots\!04}{27\!\cdots\!13}a^{3}-\frac{33\!\cdots\!17}{54\!\cdots\!26}a^{2}+\frac{13\!\cdots\!94}{27\!\cdots\!13}a-\frac{18\!\cdots\!75}{54\!\cdots\!26}$, $\frac{260}{157937209}a^{11}-\frac{3242}{157937209}a^{10}+\frac{29570}{157937209}a^{9}-\frac{221385}{157937209}a^{8}+\frac{1243840}{157937209}a^{7}-\frac{6885166}{157937209}a^{6}+\frac{25756480}{157937209}a^{5}-\frac{112467875}{157937209}a^{4}+\frac{285328360}{157937209}a^{3}-\frac{1022218817}{157937209}a^{2}+\frac{1283016620}{157937209}a-\frac{4461256592}{157937209}$ 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:  \( 3407.79685773 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 3407.79685773 \cdot 8788}{2\cdot\sqrt{25358702738605805230358528}}\cr\approx \mathstrut & 0.182957174429 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 92*x^10 - 300*x^9 + 3971*x^8 - 9660*x^7 + 97334*x^6 - 167928*x^5 + 1444284*x^4 - 1443204*x^3 + 12846660*x^2 - 4211292*x + 52322057)
 
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 - 4*x^11 + 92*x^10 - 300*x^9 + 3971*x^8 - 9660*x^7 + 97334*x^6 - 167928*x^5 + 1444284*x^4 - 1443204*x^3 + 12846660*x^2 - 4211292*x + 52322057, 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 - 4*x^11 + 92*x^10 - 300*x^9 + 3971*x^8 - 9660*x^7 + 97334*x^6 - 167928*x^5 + 1444284*x^4 - 1443204*x^3 + 12846660*x^2 - 4211292*x + 52322057);
 
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 - 4*x^11 + 92*x^10 - 300*x^9 + 3971*x^8 - 9660*x^7 + 97334*x^6 - 167928*x^5 + 1444284*x^4 - 1443204*x^3 + 12846660*x^2 - 4211292*x + 52322057);
 
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

\(\Q(\sqrt{17}) \), 3.3.169.1, 4.0.314432.2, 6.6.140320193.1

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 ${\href{/padicField/3.12.0.1}{12} }$ ${\href{/padicField/5.4.0.1}{4} }^{3}$ ${\href{/padicField/7.12.0.1}{12} }$ ${\href{/padicField/11.12.0.1}{12} }$ R R ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.12.0.1}{12} }$ ${\href{/padicField/31.4.0.1}{4} }^{3}$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{6}$ ${\href{/padicField/53.1.0.1}{1} }^{12}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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 2.6.9.7$x^{6} + 32 x^{4} + 2 x^{3} + 301 x^{2} - 58 x + 811$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 32 x^{4} + 2 x^{3} + 301 x^{2} - 58 x + 811$$2$$3$$9$$C_6$$[3]^{3}$
\(13\) Copy content Toggle raw display 13.6.4.3$x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
\(17\) 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}$