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Properties

Label 12.0.75794375168.1
Degree $12$
Signature $[0, 6]$
Discriminant $75794375168$
Root discriminant \(8.07\)
Ramified primes $2,23$
Class number $1$
Class group trivial
Galois group $(C_6\times C_2):C_2$ (as 12T15)

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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 + 9*x^10 - 12*x^9 + 14*x^8 - 19*x^7 + 30*x^6 - 39*x^5 + 37*x^4 - 26*x^3 + 14*x^2 - 5*x + 1)
 
gp: K = bnfinit(y^12 - 4*y^11 + 9*y^10 - 12*y^9 + 14*y^8 - 19*y^7 + 30*y^6 - 39*y^5 + 37*y^4 - 26*y^3 + 14*y^2 - 5*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 9*x^10 - 12*x^9 + 14*x^8 - 19*x^7 + 30*x^6 - 39*x^5 + 37*x^4 - 26*x^3 + 14*x^2 - 5*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 + 9*x^10 - 12*x^9 + 14*x^8 - 19*x^7 + 30*x^6 - 39*x^5 + 37*x^4 - 26*x^3 + 14*x^2 - 5*x + 1)
 

\( x^{12} - 4 x^{11} + 9 x^{10} - 12 x^{9} + 14 x^{8} - 19 x^{7} + 30 x^{6} - 39 x^{5} + 37 x^{4} + \cdots + 1 \) 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:   \(75794375168\) \(\medspace = 2^{9}\cdot 23^{6}\) 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:  \(8.07\)
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:  $2^{3/2}23^{1/2}\approx 13.564659966250536$
Ramified primes:   \(2\), \(23\) 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:  \(\Q(\sqrt{2}) \)
$\card{ \Aut(K/\Q) }$:  $6$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{167}a^{11}-\frac{30}{167}a^{10}-\frac{46}{167}a^{9}+\frac{15}{167}a^{8}-\frac{42}{167}a^{7}+\frac{71}{167}a^{6}+\frac{21}{167}a^{5}+\frac{83}{167}a^{4}+\frac{50}{167}a^{3}+\frac{10}{167}a^{2}-\frac{79}{167}a+\frac{45}{167}$ 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:   $\frac{291}{167}a^{11}-\frac{881}{167}a^{10}+\frac{1644}{167}a^{9}-\frac{1480}{167}a^{8}+\frac{1806}{167}a^{7}-\frac{2886}{167}a^{6}+\frac{4942}{167}a^{5}-\frac{5072}{167}a^{4}+\frac{3361}{167}a^{3}-\frac{1432}{167}a^{2}+\frac{391}{167}a+\frac{69}{167}$, $\frac{181}{167}a^{11}-\frac{754}{167}a^{10}+\frac{1694}{167}a^{9}-\frac{2295}{167}a^{8}+\frac{2585}{167}a^{7}-\frac{3515}{167}a^{6}+\frac{5471}{167}a^{5}-\frac{7355}{167}a^{4}+\frac{6879}{167}a^{3}-\frac{4703}{167}a^{2}+\frac{2234}{167}a-\frac{706}{167}$, $\frac{211}{167}a^{11}-\frac{819}{167}a^{10}+\frac{1817}{167}a^{9}-\frac{2346}{167}a^{8}+\frac{2661}{167}a^{7}-\frac{3556}{167}a^{6}+\frac{5767}{167}a^{5}-\frac{7537}{167}a^{4}+\frac{6876}{167}a^{3}-\frac{4236}{167}a^{2}+\frac{1868}{167}a-\frac{358}{167}$, $\frac{62}{167}a^{11}-\frac{357}{167}a^{10}+\frac{822}{167}a^{9}-\frac{1241}{167}a^{8}+\frac{1237}{167}a^{7}-\frac{1777}{167}a^{6}+\frac{2638}{167}a^{5}-\frac{3872}{167}a^{4}+\frac{3601}{167}a^{3}-\frac{2386}{167}a^{2}+\frac{1114}{167}a-\frac{216}{167}$, $a$ 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:  \( 1.74341488875 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1.74341488875 \cdot 1}{2\cdot\sqrt{75794375168}}\cr\approx \mathstrut & 0.194819074773 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 9*x^10 - 12*x^9 + 14*x^8 - 19*x^7 + 30*x^6 - 39*x^5 + 37*x^4 - 26*x^3 + 14*x^2 - 5*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 - 4*x^11 + 9*x^10 - 12*x^9 + 14*x^8 - 19*x^7 + 30*x^6 - 39*x^5 + 37*x^4 - 26*x^3 + 14*x^2 - 5*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 - 4*x^11 + 9*x^10 - 12*x^9 + 14*x^8 - 19*x^7 + 30*x^6 - 39*x^5 + 37*x^4 - 26*x^3 + 14*x^2 - 5*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 - 4*x^11 + 9*x^10 - 12*x^9 + 14*x^8 - 19*x^7 + 30*x^6 - 39*x^5 + 37*x^4 - 26*x^3 + 14*x^2 - 5*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

$C_3:D_4$ (as 12T15):

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 24
The 9 conjugacy class representatives for $(C_6\times C_2):C_2$
Character table for $(C_6\times C_2):C_2$

Intermediate fields

\(\Q(\sqrt{-23}) \), 3.1.23.1 x3, 4.0.4232.1, 6.0.12167.1

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 24
Degree 12 sibling: 12.2.1687248699392.3
Minimal sibling: 12.2.1687248699392.3

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{3}$ ${\href{/padicField/7.2.0.1}{2} }^{6}$ ${\href{/padicField/11.4.0.1}{4} }^{3}$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{6}$ ${\href{/padicField/19.4.0.1}{4} }^{3}$ R ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$

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 2.6.0.1$x^{6} + x^{4} + x^{3} + x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.9.3$x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$$2$$3$$9$$C_6$$[3]^{3}$
\(23\) Copy content Toggle raw display 23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$