↑

Properties

Label 12.4.1255569610725.1
Degree $12$
Signature $[4, 4]$
Discriminant $1.256\times 10^{12}$
Root discriminant \(10.19\)
Ramified primes $3,5,17,23$
Class number $1$
Class group trivial
Galois group $C_2\wr D_6$ (as 12T186)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

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

\( x^{12} - x^{11} + 2x^{10} - 3x^{9} - 4x^{8} + 3x^{7} - 3x^{6} + 3x^{5} + 2x^{4} + 8x^{3} - 16x^{2} + 8x - 1 \) 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:  $[4, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(1255569610725\) \(\medspace = 3^{3}\cdot 5^{2}\cdot 17^{2}\cdot 23^{5}\) 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:  \(10.19\)
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:  $3^{1/2}5^{1/2}17^{1/2}23^{1/2}\approx 76.58328799418317$
Ramified primes:   \(3\), \(5\), \(17\), \(23\) 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:  \(\Q(\sqrt{69}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{46199}a^{11}+\frac{11723}{46199}a^{10}-\frac{1571}{46199}a^{9}+\frac{14994}{46199}a^{8}+\frac{2457}{46199}a^{7}-\frac{22305}{46199}a^{6}-\frac{17483}{46199}a^{5}+\frac{14274}{46199}a^{4}+\frac{15600}{46199}a^{3}-\frac{7433}{46199}a^{2}-\frac{13194}{46199}a-\frac{12196}{46199}$ 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:  $7$
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:   $\frac{120}{46199}a^{11}-\frac{25409}{46199}a^{10}-\frac{3724}{46199}a^{9}-\frac{48680}{46199}a^{8}+\frac{17646}{46199}a^{7}+\frac{141539}{46199}a^{6}+\frac{73393}{46199}a^{5}+\frac{142114}{46199}a^{4}+\frac{70239}{46199}a^{3}-\frac{14179}{46199}a^{2}-\frac{243509}{46199}a+\frac{107246}{46199}$, $\frac{41886}{46199}a^{11}-\frac{19593}{46199}a^{10}+\frac{76868}{46199}a^{9}-\frac{82920}{46199}a^{8}-\frac{202266}{46199}a^{7}+\frac{15147}{46199}a^{6}-\frac{131186}{46199}a^{5}+\frac{19505}{46199}a^{4}+\frac{75342}{46199}a^{3}+\frac{366015}{46199}a^{2}-\frac{473436}{46199}a+\frac{165483}{46199}$, $\frac{4702}{46199}a^{11}+\frac{6139}{46199}a^{10}+\frac{4998}{46199}a^{9}+\frac{2114}{46199}a^{8}-\frac{43135}{46199}a^{7}-\frac{52579}{46199}a^{6}-\frac{17045}{46199}a^{5}-\frac{10799}{46199}a^{4}+\frac{33387}{46199}a^{3}+\frac{115075}{46199}a^{2}+\frac{53268}{46199}a-\frac{58832}{46199}$, $\frac{17564}{46199}a^{11}-\frac{6171}{46199}a^{10}+\frac{33958}{46199}a^{9}-\frac{25883}{46199}a^{8}-\frac{87516}{46199}a^{7}+\frac{2500}{46199}a^{6}-\frac{79057}{46199}a^{5}-\frac{13437}{46199}a^{4}+\frac{38330}{46199}a^{3}+\frac{143759}{46199}a^{2}-\frac{190028}{46199}a+\frac{14219}{46199}$, $\frac{20587}{46199}a^{11}-\frac{2175}{46199}a^{10}+\frac{43322}{46199}a^{9}-\frac{20240}{46199}a^{8}-\frac{98044}{46199}a^{7}-\frac{21174}{46199}a^{6}-\frac{124709}{46199}a^{5}-\frac{59200}{46199}a^{4}-\frac{18248}{46199}a^{3}+\frac{126514}{46199}a^{2}-\frac{159554}{46199}a+\frac{58712}{46199}$, $\frac{22266}{46199}a^{11}-\frac{32}{46199}a^{10}+\frac{38956}{46199}a^{9}-\frac{23769}{46199}a^{8}-\frac{130651}{46199}a^{7}-\frac{50079}{46199}a^{6}-\frac{96102}{46199}a^{5}-\frac{24236}{46199}a^{4}+\frac{71717}{46199}a^{3}+\frac{258834}{46199}a^{2}-\frac{136760}{46199}a-\frac{44613}{46199}$, $\frac{16826}{46199}a^{11}-\frac{18532}{46199}a^{10}+\frac{38381}{46199}a^{9}-\frac{49894}{46199}a^{8}-\frac{52822}{46199}a^{7}+\frac{62945}{46199}a^{6}-\frac{66124}{46199}a^{5}+\frac{31922}{46199}a^{4}-\frac{17118}{46199}a^{3}+\frac{85433}{46199}a^{2}-\frac{339442}{46199}a+\frac{144659}{46199}$ 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:  \( 15.2646141632 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 15.2646141632 \cdot 1}{2\cdot\sqrt{1255569610725}}\cr\approx \mathstrut & 0.169853653909 \end{aligned}\]

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

$C_2\wr D_6$ (as 12T186):

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 768
The 38 conjugacy class representatives for $C_2\wr D_6$
Character table for $C_2\wr D_6$

Intermediate fields

3.1.23.1, 6.2.44965.1

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 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 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 ${\href{/padicField/2.12.0.1}{12} }$ R R ${\href{/padicField/7.4.0.1}{4} }^{3}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ R ${\href{/padicField/19.4.0.1}{4} }^{3}$ R ${\href{/padicField/29.12.0.1}{12} }$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(3\) Copy content Toggle raw display 3.6.3.1$x^{6} + 18 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.0.1$x^{6} + 2 x^{4} + x^{2} + 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + 4 x^{2} + 4 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + 4 x^{2} + 4 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
\(17\) Copy content Toggle raw display 17.4.0.1$x^{4} + 7 x^{2} + 10 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} + 7 x^{2} + 10 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(23\) Copy content Toggle raw display 23.2.1.2$x^{2} + 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.8.4.1$x^{8} + 98 x^{6} + 38 x^{5} + 3331 x^{4} - 1634 x^{3} + 44919 x^{2} - 57494 x + 224528$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$