Properties

Label 12.0.5596214759424.1
Degree $12$
Signature $[0, 6]$
Discriminant $5.596\times 10^{12}$
Root discriminant \(11.54\)
Ramified primes $2,3,37$
Class number $1$
Class group trivial
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^10 + 9*x^8 - 9*x^6 + 20*x^4 - 7*x^2 + 1)
 
gp: K = bnfinit(y^12 - 2*y^10 + 9*y^8 - 9*y^6 + 20*y^4 - 7*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^10 + 9*x^8 - 9*x^6 + 20*x^4 - 7*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^10 + 9*x^8 - 9*x^6 + 20*x^4 - 7*x^2 + 1)
 

\( x^{12} - 2x^{10} + 9x^{8} - 9x^{6} + 20x^{4} - 7x^{2} + 1 \) 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:   \(5596214759424\) \(\medspace = 2^{12}\cdot 3^{6}\cdot 37^{4}\) 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:  \(11.54\)
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:  $2\cdot 3^{1/2}37^{2/3}\approx 38.464353655440554$
Ramified primes:   \(2\), \(3\), \(37\) 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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{77}a^{10}+\frac{4}{77}a^{8}+\frac{3}{7}a^{6}+\frac{5}{11}a^{4}-\frac{1}{77}a^{2}-\frac{13}{77}$, $\frac{1}{77}a^{11}+\frac{4}{77}a^{9}+\frac{3}{7}a^{7}+\frac{5}{11}a^{5}-\frac{1}{77}a^{3}-\frac{13}{77}a$ 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

Trivial group, which has order $1$

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:   \( \frac{51}{77} a^{11} - \frac{104}{77} a^{9} + \frac{41}{7} a^{7} - \frac{64}{11} a^{5} + \frac{950}{77} a^{3} - \frac{278}{77} a \)  (order $12$) 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:   $\frac{26}{77}a^{10}-\frac{50}{77}a^{8}+\frac{22}{7}a^{6}-\frac{35}{11}a^{4}+\frac{590}{77}a^{2}-\frac{184}{77}$, $\frac{15}{77}a^{10}-\frac{17}{77}a^{8}+\frac{10}{7}a^{6}-\frac{2}{11}a^{4}+\frac{216}{77}a^{2}+\frac{113}{77}$, $\frac{51}{77}a^{11}-\frac{104}{77}a^{9}+\frac{41}{7}a^{7}-\frac{64}{11}a^{5}+\frac{950}{77}a^{3}-\frac{278}{77}a-1$, $\frac{39}{77}a^{11}-\frac{26}{77}a^{10}-\frac{75}{77}a^{9}+\frac{50}{77}a^{8}+\frac{33}{7}a^{7}-\frac{22}{7}a^{6}-\frac{47}{11}a^{5}+\frac{35}{11}a^{4}+\frac{808}{77}a^{3}-\frac{590}{77}a^{2}-\frac{122}{77}a+\frac{107}{77}$, $\frac{79}{77}a^{11}-\frac{40}{77}a^{10}-\frac{146}{77}a^{9}+\frac{71}{77}a^{8}+\frac{62}{7}a^{7}-\frac{29}{7}a^{6}-\frac{89}{11}a^{5}+\frac{42}{11}a^{4}+\frac{1461}{77}a^{3}-\frac{653}{77}a^{2}-\frac{411}{77}a+\frac{212}{77}$ Copy content Toggle raw display
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:  \( 123.785869620809 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 123.785869620809 \cdot 1}{12\cdot\sqrt{5596214759424}}\cr\approx \mathstrut & 0.268300727237975 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^10 + 9*x^8 - 9*x^6 + 20*x^4 - 7*x^2 + 1)
 
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^10 + 9*x^8 - 9*x^6 + 20*x^4 - 7*x^2 + 1, 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^10 + 9*x^8 - 9*x^6 + 20*x^4 - 7*x^2 + 1);
 
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^10 + 9*x^8 - 9*x^6 + 20*x^4 - 7*x^2 + 1);
 
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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{12})\), 6.0.36963.1

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: deg 36
Degree 18 siblings: 18.6.33966586417892403297890598912.1, 18.0.1258021719181200122144096256.1
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 R ${\href{/padicField/5.6.0.1}{6} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ ${\href{/padicField/11.2.0.1}{2} }^{6}$ ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{6}$ ${\href{/padicField/29.2.0.1}{2} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ R ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{6}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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 2.12.12.26$x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
\(3\) Copy content Toggle raw display 3.12.6.2$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
\(37\) Copy content Toggle raw display $\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
37.3.2.1$x^{3} + 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} + 37$$3$$1$$2$$C_3$$[\ ]_{3}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.12.2t1.a.a$1$ $ 2^{2} \cdot 3 $ \(\Q(\sqrt{3}) \) $C_2$ (as 2T1) $1$ $1$
* 1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
1.148.6t1.b.a$1$ $ 2^{2} \cdot 37 $ 6.0.119946304.1 $C_6$ (as 6T1) $0$ $-1$
1.444.6t1.b.a$1$ $ 2^{2} \cdot 3 \cdot 37 $ 6.6.3238550208.1 $C_6$ (as 6T1) $0$ $1$
1.444.6t1.b.b$1$ $ 2^{2} \cdot 3 \cdot 37 $ 6.6.3238550208.1 $C_6$ (as 6T1) $0$ $1$
1.111.6t1.b.a$1$ $ 3 \cdot 37 $ 6.0.50602347.1 $C_6$ (as 6T1) $0$ $-1$
1.111.6t1.b.b$1$ $ 3 \cdot 37 $ 6.0.50602347.1 $C_6$ (as 6T1) $0$ $-1$
1.148.6t1.b.b$1$ $ 2^{2} \cdot 37 $ 6.0.119946304.1 $C_6$ (as 6T1) $0$ $-1$
1.37.3t1.a.a$1$ $ 37 $ 3.3.1369.1 $C_3$ (as 3T1) $0$ $1$
1.37.3t1.a.b$1$ $ 37 $ 3.3.1369.1 $C_3$ (as 3T1) $0$ $1$
2.4107.3t2.a.a$2$ $ 3 \cdot 37^{2}$ 3.1.4107.1 $S_3$ (as 3T2) $1$ $0$
2.65712.6t3.b.a$2$ $ 2^{4} \cdot 3 \cdot 37^{2}$ 6.2.3238550208.6 $D_{6}$ (as 6T3) $1$ $0$
* 2.111.6t5.a.a$2$ $ 3 \cdot 37 $ 6.0.36963.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.1776.12t18.b.a$2$ $ 2^{4} \cdot 3 \cdot 37 $ 12.0.5596214759424.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.111.6t5.a.b$2$ $ 3 \cdot 37 $ 6.0.36963.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.1776.12t18.b.b$2$ $ 2^{4} \cdot 3 \cdot 37 $ 12.0.5596214759424.1 $C_6\times S_3$ (as 12T18) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.