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Properties

Label 12.0.179...968.1
Degree $12$
Signature $[0, 6]$
Discriminant $1.792\times 10^{23}$
Root discriminant \(86.65\)
Ramified primes $2,7,17$
Class number $2164$ (GRH)
Class group [2, 1082] (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 - 4*x^11 + 100*x^10 - 316*x^9 + 4307*x^8 - 10660*x^7 + 101974*x^6 - 189064*x^5 + 1400852*x^4 - 1767732*x^3 + 10735044*x^2 - 7114100*x + 35851369)
 
gp: K = bnfinit(y^12 - 4*y^11 + 100*y^10 - 316*y^9 + 4307*y^8 - 10660*y^7 + 101974*y^6 - 189064*y^5 + 1400852*y^4 - 1767732*y^3 + 10735044*y^2 - 7114100*y + 35851369, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 100*x^10 - 316*x^9 + 4307*x^8 - 10660*x^7 + 101974*x^6 - 189064*x^5 + 1400852*x^4 - 1767732*x^3 + 10735044*x^2 - 7114100*x + 35851369);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 100*x^10 - 316*x^9 + 4307*x^8 - 10660*x^7 + 101974*x^6 - 189064*x^5 + 1400852*x^4 - 1767732*x^3 + 10735044*x^2 - 7114100*x + 35851369)
 

\( x^{12} - 4 x^{11} + 100 x^{10} - 316 x^{9} + 4307 x^{8} - 10660 x^{7} + 101974 x^{6} - 189064 x^{5} + \cdots + 35851369 \) 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:   \(179210946875957470035968\) \(\medspace = 2^{18}\cdot 7^{8}\cdot 17^{9}\) 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:  \(86.65\)
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:  $2^{3/2}7^{2/3}17^{3/4}\approx 86.6523565155333$
Ramified primes:   \(2\), \(7\), \(17\) 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:  \(952=2^{3}\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{952}(1,·)$, $\chi_{952}(387,·)$, $\chi_{952}(667,·)$, $\chi_{952}(849,·)$, $\chi_{952}(795,·)$, $\chi_{952}(169,·)$, $\chi_{952}(939,·)$, $\chi_{952}(305,·)$, $\chi_{952}(659,·)$, $\chi_{952}(681,·)$, $\chi_{952}(137,·)$, $\chi_{952}(123,·)$$\rbrace$
This is a CM field.
Reflex fields:  4.0.314432.2$^{2}$, 12.0.179210946875957470035968.1$^{30}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{167536036}a^{10}+\frac{10275849}{83768018}a^{9}-\frac{3951875}{83768018}a^{8}+\frac{41624113}{167536036}a^{7}-\frac{992785}{41884009}a^{6}+\frac{62534063}{167536036}a^{5}-\frac{13434177}{83768018}a^{4}+\frac{6448224}{41884009}a^{3}+\frac{69814037}{167536036}a^{2}+\frac{76700853}{167536036}a-\frac{20667965}{167536036}$, $\frac{1}{20\!\cdots\!72}a^{11}+\frac{40045864735}{20\!\cdots\!72}a^{10}-\frac{28\!\cdots\!75}{52\!\cdots\!93}a^{9}-\frac{80\!\cdots\!25}{20\!\cdots\!72}a^{8}+\frac{25\!\cdots\!17}{20\!\cdots\!72}a^{7}-\frac{19\!\cdots\!63}{20\!\cdots\!72}a^{6}+\frac{57\!\cdots\!41}{20\!\cdots\!72}a^{5}-\frac{59\!\cdots\!85}{52\!\cdots\!93}a^{4}-\frac{98\!\cdots\!63}{20\!\cdots\!72}a^{3}+\frac{24\!\cdots\!85}{10\!\cdots\!86}a^{2}+\frac{71\!\cdots\!71}{52\!\cdots\!93}a-\frac{96\!\cdots\!47}{20\!\cdots\!72}$ 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

$C_{2}\times C_{1082}$, which has order $2164$ (assuming GRH)

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

Relative class number: $2164$ (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:   $\frac{3949912542168}{52\!\cdots\!93}a^{11}-\frac{15722063860456}{52\!\cdots\!93}a^{10}+\frac{297653402666344}{52\!\cdots\!93}a^{9}-\frac{949805556041429}{52\!\cdots\!93}a^{8}+\frac{92\!\cdots\!12}{52\!\cdots\!93}a^{7}-\frac{24\!\cdots\!72}{52\!\cdots\!93}a^{6}+\frac{14\!\cdots\!40}{52\!\cdots\!93}a^{5}-\frac{30\!\cdots\!84}{52\!\cdots\!93}a^{4}+\frac{94\!\cdots\!12}{52\!\cdots\!93}a^{3}-\frac{17\!\cdots\!04}{52\!\cdots\!93}a^{2}+\frac{16\!\cdots\!88}{52\!\cdots\!93}a-\frac{81\!\cdots\!59}{52\!\cdots\!93}$, $\frac{3949912542168}{52\!\cdots\!93}a^{11}-\frac{15722063860456}{52\!\cdots\!93}a^{10}+\frac{297653402666344}{52\!\cdots\!93}a^{9}-\frac{949805556041429}{52\!\cdots\!93}a^{8}+\frac{92\!\cdots\!12}{52\!\cdots\!93}a^{7}-\frac{24\!\cdots\!72}{52\!\cdots\!93}a^{6}+\frac{14\!\cdots\!40}{52\!\cdots\!93}a^{5}-\frac{30\!\cdots\!84}{52\!\cdots\!93}a^{4}+\frac{94\!\cdots\!12}{52\!\cdots\!93}a^{3}-\frac{17\!\cdots\!04}{52\!\cdots\!93}a^{2}+\frac{16\!\cdots\!88}{52\!\cdots\!93}a-\frac{29\!\cdots\!66}{52\!\cdots\!93}$, $\frac{2505229883270}{52\!\cdots\!93}a^{11}-\frac{78156107983495}{52\!\cdots\!93}a^{10}+\frac{381425283913983}{52\!\cdots\!93}a^{9}-\frac{10\!\cdots\!23}{10\!\cdots\!86}a^{8}+\frac{16\!\cdots\!56}{52\!\cdots\!93}a^{7}-\frac{16\!\cdots\!37}{52\!\cdots\!93}a^{6}+\frac{30\!\cdots\!30}{52\!\cdots\!93}a^{5}-\frac{48\!\cdots\!05}{10\!\cdots\!86}a^{4}+\frac{25\!\cdots\!40}{52\!\cdots\!93}a^{3}-\frac{37\!\cdots\!33}{10\!\cdots\!86}a^{2}+\frac{77\!\cdots\!10}{52\!\cdots\!93}a-\frac{13\!\cdots\!95}{10\!\cdots\!86}$, $\frac{5092803665086}{52\!\cdots\!93}a^{11}+\frac{28683463264457}{52\!\cdots\!93}a^{10}+\frac{257727004553937}{52\!\cdots\!93}a^{9}+\frac{34\!\cdots\!51}{10\!\cdots\!86}a^{8}+\frac{68\!\cdots\!40}{52\!\cdots\!93}a^{7}+\frac{40\!\cdots\!99}{52\!\cdots\!93}a^{6}+\frac{11\!\cdots\!22}{52\!\cdots\!93}a^{5}+\frac{55\!\cdots\!51}{10\!\cdots\!86}a^{4}+\frac{13\!\cdots\!76}{52\!\cdots\!93}a^{3}-\frac{33\!\cdots\!31}{10\!\cdots\!86}a^{2}+\frac{77\!\cdots\!78}{52\!\cdots\!93}a-\frac{40\!\cdots\!93}{10\!\cdots\!86}$, $\frac{149841796626}{52\!\cdots\!93}a^{11}-\frac{51606544465274}{52\!\cdots\!93}a^{10}+\frac{108408951570723}{52\!\cdots\!93}a^{9}-\frac{38\!\cdots\!36}{52\!\cdots\!93}a^{8}+\frac{59\!\cdots\!10}{52\!\cdots\!93}a^{7}-\frac{26\!\cdots\!95}{10\!\cdots\!86}a^{6}+\frac{16\!\cdots\!66}{52\!\cdots\!93}a^{5}-\frac{50\!\cdots\!47}{10\!\cdots\!86}a^{4}+\frac{19\!\cdots\!60}{52\!\cdots\!93}a^{3}-\frac{26\!\cdots\!24}{52\!\cdots\!93}a^{2}+\frac{10\!\cdots\!97}{52\!\cdots\!93}a-\frac{24\!\cdots\!61}{10\!\cdots\!86}$ 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:  \( 1059.54542703 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1059.54542703 \cdot 2164}{2\cdot\sqrt{179210946875957470035968}}\cr\approx \mathstrut & 0.166626389914 \end{aligned}\] (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 + 100*x^10 - 316*x^9 + 4307*x^8 - 10660*x^7 + 101974*x^6 - 189064*x^5 + 1400852*x^4 - 1767732*x^3 + 10735044*x^2 - 7114100*x + 35851369)
 
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 + 100*x^10 - 316*x^9 + 4307*x^8 - 10660*x^7 + 101974*x^6 - 189064*x^5 + 1400852*x^4 - 1767732*x^3 + 10735044*x^2 - 7114100*x + 35851369, 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 + 100*x^10 - 316*x^9 + 4307*x^8 - 10660*x^7 + 101974*x^6 - 189064*x^5 + 1400852*x^4 - 1767732*x^3 + 10735044*x^2 - 7114100*x + 35851369);
 
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 + 100*x^10 - 316*x^9 + 4307*x^8 - 10660*x^7 + 101974*x^6 - 189064*x^5 + 1400852*x^4 - 1767732*x^3 + 10735044*x^2 - 7114100*x + 35851369);
 
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

\(\Q(\sqrt{17}) \), \(\Q(\zeta_{7})^+\), 4.0.314432.2, 6.6.11796113.1

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 ${\href{/padicField/3.12.0.1}{12} }$ ${\href{/padicField/5.12.0.1}{12} }$ R ${\href{/padicField/11.12.0.1}{12} }$ ${\href{/padicField/13.2.0.1}{2} }^{6}$ R ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.4.0.1}{4} }^{3}$ ${\href{/padicField/31.12.0.1}{12} }$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.4.0.1}{4} }^{3}$ ${\href{/padicField/43.2.0.1}{2} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.3.0.1}{3} }^{4}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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}$
\(7\) Copy content Toggle raw display 7.12.8.1$x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$$3$$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}$