Properties

Label 12.6.30755756000000.1
Degree $12$
Signature $[6, 3]$
Discriminant $-3.076\times 10^{13}$
Root discriminant \(13.30\)
Ramified primes $2,5,19,59$
Class number $1$
Class group trivial
Galois group $A_4^2:D_4$ (as 12T208)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 6*x^10 + 12*x^9 - x^8 - 26*x^7 + 41*x^6 - 26*x^5 - x^4 + 12*x^3 - 6*x^2 - x + 1)
 
gp: K = bnfinit(y^12 - y^11 - 6*y^10 + 12*y^9 - y^8 - 26*y^7 + 41*y^6 - 26*y^5 - y^4 + 12*y^3 - 6*y^2 - y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 6*x^10 + 12*x^9 - x^8 - 26*x^7 + 41*x^6 - 26*x^5 - x^4 + 12*x^3 - 6*x^2 - x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - x^11 - 6*x^10 + 12*x^9 - x^8 - 26*x^7 + 41*x^6 - 26*x^5 - x^4 + 12*x^3 - 6*x^2 - x + 1)
 

\( x^{12} - x^{11} - 6x^{10} + 12x^{9} - x^{8} - 26x^{7} + 41x^{6} - 26x^{5} - x^{4} + 12x^{3} - 6x^{2} - x + 1 \) 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:  $[6, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-30755756000000\) \(\medspace = -\,2^{8}\cdot 5^{6}\cdot 19^{4}\cdot 59\) 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:  \(13.30\)
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:  $2^{2/3}5^{1/2}19^{2/3}59^{1/2}\approx 194.13331581729184$
Ramified primes:   \(2\), \(5\), \(19\), \(59\) 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:  \(\Q(\sqrt{-59}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Yes
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:  $8$
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:   $7a^{11}-10a^{10}-38a^{9}+100a^{8}-48a^{7}-161a^{6}+352a^{5}-329a^{4}+136a^{3}+18a^{2}-45a+12$, $3a^{11}-6a^{10}-15a^{9}+53a^{8}-38a^{7}-69a^{6}+189a^{5}-200a^{4}+100a^{3}-a^{2}-26a+10$, $2a^{11}-2a^{10}-11a^{9}+23a^{8}-8a^{7}-40a^{6}+81a^{5}-78a^{4}+38a^{3}-a^{2}-10a+4$, $a^{11}-a^{10}-6a^{9}+12a^{8}-a^{7}-26a^{6}+41a^{5}-26a^{4}-a^{3}+12a^{2}-6a-1$, $6a^{11}-8a^{10}-33a^{9}+82a^{8}-35a^{7}-136a^{6}+286a^{5}-264a^{4}+108a^{3}+14a^{2}-35a+10$, $a-1$, $3a^{11}-7a^{10}-14a^{9}+59a^{8}-50a^{7}-68a^{6}+215a^{5}-240a^{4}+126a^{3}-3a^{2}-35a+13$, $6a^{11}-9a^{10}-32a^{9}+88a^{8}-47a^{7}-135a^{6}+312a^{5}-304a^{4}+134a^{3}+12a^{2}-44a+14$ 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:  \( 127.148219868 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{3}\cdot 127.148219868 \cdot 1}{2\cdot\sqrt{30755756000000}}\cr\approx \mathstrut & 0.181985448885 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 6*x^10 + 12*x^9 - x^8 - 26*x^7 + 41*x^6 - 26*x^5 - x^4 + 12*x^3 - 6*x^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 - x^11 - 6*x^10 + 12*x^9 - x^8 - 26*x^7 + 41*x^6 - 26*x^5 - x^4 + 12*x^3 - 6*x^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 - x^11 - 6*x^10 + 12*x^9 - x^8 - 26*x^7 + 41*x^6 - 26*x^5 - x^4 + 12*x^3 - 6*x^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 - x^11 - 6*x^10 + 12*x^9 - x^8 - 26*x^7 + 41*x^6 - 26*x^5 - x^4 + 12*x^3 - 6*x^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

$A_4^2:D_4$ (as 12T208):

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 1152
The 44 conjugacy class representatives for $A_4^2:D_4$
Character table for $A_4^2:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 6.6.722000.1

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 12 sibling: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: data not computed
Degree 36 siblings: data not computed
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 ${\href{/padicField/3.6.0.1}{6} }^{2}$ R ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.12.0.1}{12} }$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ R ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ R

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 2.12.8.2$x^{12} - 8 x^{3} + 16$$3$$4$$8$$C_3\times (C_3 : C_4)$$[\ ]_{3}^{12}$
\(5\) Copy content Toggle raw display 5.6.3.1$x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(19\) Copy content Toggle raw display 19.6.0.1$x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
19.6.4.2$x^{6} - 342 x^{3} + 722$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
\(59\) Copy content Toggle raw display $\Q_{59}$$x + 57$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 57$$1$$1$$0$Trivial$[\ ]$
59.2.0.1$x^{2} + 58 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.1.1$x^{2} + 118$$2$$1$$1$$C_2$$[\ ]_{2}$
59.3.0.1$x^{3} + 5 x + 57$$1$$3$$0$$C_3$$[\ ]^{3}$
59.3.0.1$x^{3} + 5 x + 57$$1$$3$$0$$C_3$$[\ ]^{3}$