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Properties

Label 12.4.350284500000000.1
Degree $12$
Signature $[4, 4]$
Discriminant $3.503\times 10^{14}$
Root discriminant \(16.29\)
Ramified primes $2,3,5,31$
Class number $1$
Class group trivial
Galois group $C_3^3:(C_4\times S_3)$ (as 12T170)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

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

\( x^{12} - 3x^{11} + x^{10} + x^{9} - x^{8} + 8x^{7} - 11x^{6} + 14x^{5} - 9x^{4} + 5x^{3} - 10x^{2} - 5 \) 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:   \(350284500000000\) \(\medspace = 2^{8}\cdot 3^{6}\cdot 5^{9}\cdot 31^{2}\) 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:  \(16.29\)
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:  $2^{2/3}3^{1/2}5^{3/4}31^{2/3}\approx 90.7226909805036$
Ramified primes:   \(2\), \(3\), \(5\), \(31\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{12057}a^{11}-\frac{1165}{12057}a^{10}-\frac{4691}{12057}a^{9}-\frac{2840}{12057}a^{8}+\frac{160}{4019}a^{7}+\frac{889}{12057}a^{6}+\frac{1291}{4019}a^{5}+\frac{868}{12057}a^{4}+\frac{4163}{12057}a^{3}+\frac{1475}{12057}a^{2}-\frac{5885}{12057}a-\frac{656}{4019}$ 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:   $\frac{821}{4019}a^{11}-\frac{7867}{12057}a^{10}+\frac{4711}{12057}a^{9}+\frac{2159}{12057}a^{8}-\frac{7384}{12057}a^{7}+\frac{6449}{4019}a^{6}-\frac{30040}{12057}a^{5}+\frac{13322}{4019}a^{4}-\frac{27133}{12057}a^{3}+\frac{11806}{12057}a^{2}-\frac{6260}{12057}a+\frac{15806}{12057}$, $\frac{821}{4019}a^{11}-\frac{7867}{12057}a^{10}+\frac{4711}{12057}a^{9}+\frac{2159}{12057}a^{8}-\frac{7384}{12057}a^{7}+\frac{6449}{4019}a^{6}-\frac{30040}{12057}a^{5}+\frac{13322}{4019}a^{4}-\frac{27133}{12057}a^{3}+\frac{11806}{12057}a^{2}-\frac{18317}{12057}a+\frac{15806}{12057}$, $\frac{538}{4019}a^{11}-\frac{7456}{12057}a^{10}+\frac{8560}{12057}a^{9}+\frac{1919}{12057}a^{8}-\frac{4966}{12057}a^{7}+\frac{4040}{4019}a^{6}-\frac{38713}{12057}a^{5}+\frac{16856}{4019}a^{4}-\frac{40876}{12057}a^{3}+\frac{25516}{12057}a^{2}-\frac{25607}{12057}a+\frac{22772}{12057}$, $\frac{1322}{12057}a^{11}-\frac{1624}{4019}a^{10}+\frac{1278}{4019}a^{9}-\frac{734}{12057}a^{8}-\frac{442}{12057}a^{7}+\frac{17786}{12057}a^{6}-\frac{24214}{12057}a^{5}+\frac{26195}{12057}a^{4}-\frac{8886}{4019}a^{3}+\frac{4754}{12057}a^{2}-\frac{6427}{4019}a-\frac{5422}{12057}$, $\frac{1292}{12057}a^{11}-\frac{2074}{12057}a^{10}-\frac{4139}{12057}a^{9}+\frac{1362}{4019}a^{8}+\frac{1234}{12057}a^{7}+\frac{3173}{12057}a^{6}-\frac{3758}{12057}a^{5}+\frac{155}{12057}a^{4}+\frac{9212}{12057}a^{3}+\frac{1571}{4019}a^{2}-\frac{3491}{12057}a-\frac{2648}{12057}$, $\frac{840}{4019}a^{11}-\frac{9968}{12057}a^{10}+\frac{10616}{12057}a^{9}+\frac{1039}{12057}a^{8}-\frac{12176}{12057}a^{7}+\frac{7264}{4019}a^{6}-\frac{46400}{12057}a^{5}+\frac{21776}{4019}a^{4}-\frac{39020}{12057}a^{3}+\frac{19520}{12057}a^{2}-\frac{8128}{12057}a+\frac{28219}{12057}$, $\frac{538}{4019}a^{11}-\frac{7456}{12057}a^{10}+\frac{8560}{12057}a^{9}+\frac{1919}{12057}a^{8}-\frac{4966}{12057}a^{7}+\frac{4040}{4019}a^{6}-\frac{38713}{12057}a^{5}+\frac{16856}{4019}a^{4}-\frac{40876}{12057}a^{3}+\frac{13459}{12057}a^{2}-\frac{13550}{12057}a+\frac{22772}{12057}$ 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:  \( 426.245519832 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 426.245519832 \cdot 1}{2\cdot\sqrt{350284500000000}}\cr\approx \mathstrut & 0.283961085391 \end{aligned}\]

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

$C_3^3:(C_4\times S_3)$ (as 12T170):

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 648
The 30 conjugacy class representatives for $C_3^3:(C_4\times S_3)$
Character table for $C_3^3:(C_4\times S_3)$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\)

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 18 siblings: data not computed
Degree 24 siblings: 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 R R ${\href{/padicField/7.4.0.1}{4} }^{3}$ ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.12.0.1}{12} }$ ${\href{/padicField/17.12.0.1}{12} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ ${\href{/padicField/23.4.0.1}{4} }^{3}$ ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ R ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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
\(2\) 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}$
\(3\) Copy content Toggle raw display 3.4.2.2$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(5\) Copy content Toggle raw display 5.4.3.1$x^{4} + 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
\(31\) 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$Trivial$[\ ]$
31.3.0.1$x^{3} + x + 28$$1$$3$$0$$C_3$$[\ ]^{3}$
31.3.0.1$x^{3} + x + 28$$1$$3$$0$$C_3$$[\ ]^{3}$
31.3.2.1$x^{3} + 31$$3$$1$$2$$C_3$$[\ ]_{3}$