Properties

Label 12.0.1636073786281.1
Degree $12$
Signature $[0, 6]$
Discriminant $1.636\times 10^{12}$
Root discriminant \(10.42\)
Ramified primes $11,31$
Class number $1$
Class group trivial
Galois group $C_2 \times S_4$ (as 12T23)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 7*x^10 - 5*x^9 + 20*x^8 + 21*x^7 - 16*x^6 - 22*x^5 + 10*x^4 + 11*x^3 + 7*x^2 + 2*x + 1)
 
gp: K = bnfinit(y^12 - 7*y^10 - 5*y^9 + 20*y^8 + 21*y^7 - 16*y^6 - 22*y^5 + 10*y^4 + 11*y^3 + 7*y^2 + 2*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 7*x^10 - 5*x^9 + 20*x^8 + 21*x^7 - 16*x^6 - 22*x^5 + 10*x^4 + 11*x^3 + 7*x^2 + 2*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 7*x^10 - 5*x^9 + 20*x^8 + 21*x^7 - 16*x^6 - 22*x^5 + 10*x^4 + 11*x^3 + 7*x^2 + 2*x + 1)
 

\( x^{12} - 7x^{10} - 5x^{9} + 20x^{8} + 21x^{7} - 16x^{6} - 22x^{5} + 10x^{4} + 11x^{3} + 7x^{2} + 2x + 1 \) 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:   \(1636073786281\) \(\medspace = 11^{6}\cdot 31^{4}\) 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:  \(10.42\)
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:  $11^{1/2}31^{1/2}\approx 18.466185312619388$
Ramified primes:   \(11\), \(31\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5}a^{9}-\frac{1}{5}a^{7}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{1}{5}a^{2}-\frac{2}{5}$, $\frac{1}{5}a^{10}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}+\frac{2}{5}a^{4}+\frac{1}{5}a^{3}-\frac{2}{5}a$, $\frac{1}{8235}a^{11}+\frac{677}{8235}a^{10}+\frac{152}{2745}a^{9}-\frac{929}{8235}a^{8}+\frac{1888}{8235}a^{7}+\frac{25}{1647}a^{6}+\frac{68}{915}a^{5}+\frac{2552}{8235}a^{4}+\frac{11}{8235}a^{3}+\frac{58}{549}a^{2}-\frac{629}{8235}a+\frac{4036}{8235}$ 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:   \( -1 \)  (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:   $\frac{1168}{8235}a^{11}+\frac{176}{8235}a^{10}-\frac{617}{549}a^{9}-\frac{6287}{8235}a^{8}+\frac{29497}{8235}a^{7}+\frac{27416}{8235}a^{6}-\frac{4024}{915}a^{5}-\frac{31627}{8235}a^{4}+\frac{32612}{8235}a^{3}+\frac{4379}{2745}a^{2}-\frac{9992}{8235}a+\frac{334}{8235}$, $\frac{212}{1647}a^{11}-\frac{472}{8235}a^{10}-\frac{2482}{2745}a^{9}-\frac{3128}{8235}a^{8}+\frac{4976}{1647}a^{7}+\frac{20504}{8235}a^{6}-\frac{3499}{915}a^{5}-\frac{7427}{1647}a^{4}+\frac{24836}{8235}a^{3}+\frac{9293}{2745}a^{2}-\frac{4646}{8235}a-\frac{2393}{8235}$, $\frac{449}{8235}a^{11}-\frac{722}{8235}a^{10}-\frac{926}{2745}a^{9}+\frac{2864}{8235}a^{8}+\frac{9389}{8235}a^{7}-\frac{6461}{8235}a^{6}-\frac{262}{183}a^{5}+\frac{7771}{8235}a^{4}+\frac{1976}{1647}a^{3}-\frac{2099}{2745}a^{2}-\frac{2431}{8235}a-\frac{4477}{8235}$, $\frac{274}{1647}a^{11}-\frac{1418}{8235}a^{10}-\frac{2576}{2745}a^{9}+\frac{2048}{8235}a^{8}+\frac{20534}{8235}a^{7}+\frac{1609}{8235}a^{6}-\frac{217}{183}a^{5}+\frac{1306}{8235}a^{4}-\frac{4694}{8235}a^{3}-\frac{176}{2745}a^{2}+\frac{16121}{8235}a+\frac{341}{8235}$, $\frac{659}{2745}a^{11}-\frac{743}{2745}a^{10}-\frac{1397}{915}a^{9}+\frac{424}{549}a^{8}+\frac{13334}{2745}a^{7}-\frac{3269}{2745}a^{6}-\frac{1792}{305}a^{5}+\frac{5671}{2745}a^{4}+\frac{13288}{2745}a^{3}-\frac{391}{183}a^{2}+\frac{4376}{2745}a-\frac{181}{2745}$ 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:  \( 10.2890687023 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 10.2890687023 \cdot 1}{2\cdot\sqrt{1636073786281}}\cr\approx \mathstrut & 0.247470729444 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 7*x^10 - 5*x^9 + 20*x^8 + 21*x^7 - 16*x^6 - 22*x^5 + 10*x^4 + 11*x^3 + 7*x^2 + 2*x + 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 - 7*x^10 - 5*x^9 + 20*x^8 + 21*x^7 - 16*x^6 - 22*x^5 + 10*x^4 + 11*x^3 + 7*x^2 + 2*x + 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 - 7*x^10 - 5*x^9 + 20*x^8 + 21*x^7 - 16*x^6 - 22*x^5 + 10*x^4 + 11*x^3 + 7*x^2 + 2*x + 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 - 7*x^10 - 5*x^9 + 20*x^8 + 21*x^7 - 16*x^6 - 22*x^5 + 10*x^4 + 11*x^3 + 7*x^2 + 2*x + 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

$C_2\times S_4$ (as 12T23):

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 48
The 10 conjugacy class representatives for $C_2 \times S_4$
Character table for $C_2 \times S_4$

Intermediate fields

\(\Q(\sqrt{-11}) \), 3.1.31.1, 6.0.1279091.1, 6.0.10571.1, 6.2.116281.1

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

Degree 6 siblings: 6.2.327701.1, 6.0.10571.1
Degree 8 siblings: 8.4.13521270961.1, 8.0.14070001.1
Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed
Minimal sibling: 6.0.10571.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }^{2}$ ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ ${\href{/padicField/5.3.0.1}{3} }^{4}$ ${\href{/padicField/7.6.0.1}{6} }^{2}$ R ${\href{/padicField/13.2.0.1}{2} }^{6}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ R ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.3.0.1}{3} }^{4}$

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
\(11\) Copy content Toggle raw display 11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.8.4.1$x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(31\) Copy content Toggle raw display $\Q_{31}$$x + 28$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 28$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 28$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 28$$1$$1$$0$Trivial$[\ ]$
31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$