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Properties

Label 12.4.972292630401.1
Degree $12$
Signature $[4, 4]$
Discriminant $972292630401$
Root discriminant \(9.98\)
Ramified primes $3,331$
Class number $1$
Class group trivial
Galois group $C_4^2:S_3$ (as 12T62)

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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 + 5*x^10 + 3*x^9 - 20*x^8 + 30*x^7 - 12*x^6 - 24*x^5 + 43*x^4 - 30*x^3 + 8*x^2 + 2*x - 1)
 
gp: K = bnfinit(y^12 - 4*y^11 + 5*y^10 + 3*y^9 - 20*y^8 + 30*y^7 - 12*y^6 - 24*y^5 + 43*y^4 - 30*y^3 + 8*y^2 + 2*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 5*x^10 + 3*x^9 - 20*x^8 + 30*x^7 - 12*x^6 - 24*x^5 + 43*x^4 - 30*x^3 + 8*x^2 + 2*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 + 5*x^10 + 3*x^9 - 20*x^8 + 30*x^7 - 12*x^6 - 24*x^5 + 43*x^4 - 30*x^3 + 8*x^2 + 2*x - 1)
 

\( x^{12} - 4 x^{11} + 5 x^{10} + 3 x^{9} - 20 x^{8} + 30 x^{7} - 12 x^{6} - 24 x^{5} + 43 x^{4} - 30 x^{3} + \cdots - 1 \) 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:  $[4, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(972292630401\) \(\medspace = 3^{4}\cdot 331^{4}\) 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:  \(9.98\)
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:  $3^{1/2}331^{1/2}\approx 31.51190251317746$
Ramified primes:   \(3\), \(331\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{557}a^{11}+\frac{272}{557}a^{10}-\frac{118}{557}a^{9}-\frac{259}{557}a^{8}-\frac{208}{557}a^{7}-\frac{7}{557}a^{6}-\frac{273}{557}a^{5}-\frac{177}{557}a^{4}+\frac{207}{557}a^{3}-\frac{269}{557}a^{2}-\frac{155}{557}a+\frac{111}{557}$ 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:   $\frac{753}{557}a^{11}-\frac{2388}{557}a^{10}+\frac{1937}{557}a^{9}+\frac{3265}{557}a^{8}-\frac{11804}{557}a^{7}+\frac{13667}{557}a^{6}-\frac{593}{557}a^{5}-\frac{15197}{557}a^{4}+\frac{19963}{557}a^{3}-\frac{10392}{557}a^{2}+\frac{1369}{557}a+\frac{1147}{557}$, $\frac{521}{557}a^{11}-\frac{1994}{557}a^{10}+\frac{2020}{557}a^{9}+\frac{2640}{557}a^{8}-\frac{10336}{557}a^{7}+\frac{12506}{557}a^{6}-\frac{755}{557}a^{5}-\frac{15351}{557}a^{4}+\frac{18170}{557}a^{3}-\frac{8140}{557}a^{2}-\frac{547}{557}a+\frac{1574}{557}$, $\frac{132}{557}a^{11}-\frac{858}{557}a^{10}+\frac{1691}{557}a^{9}-\frac{211}{557}a^{8}-\frac{4619}{557}a^{7}+\frac{9102}{557}a^{6}-\frac{5958}{557}a^{5}-\frac{5540}{557}a^{4}+\frac{13399}{557}a^{3}-\frac{9886}{557}a^{2}+\frac{1820}{557}a+\frac{727}{557}$, $\frac{604}{557}a^{11}-\frac{2255}{557}a^{10}+\frac{2252}{557}a^{9}+\frac{2866}{557}a^{8}-\frac{11447}{557}a^{7}+\frac{14153}{557}a^{6}-\frac{1691}{557}a^{5}-\frac{16117}{557}a^{4}+\frac{20312}{557}a^{3}-\frac{10415}{557}a^{2}+\frac{1070}{557}a+\frac{1318}{557}$, $\frac{129}{557}a^{11}-\frac{560}{557}a^{10}+\frac{931}{557}a^{9}+\frac{9}{557}a^{8}-\frac{2881}{557}a^{7}+\frac{5224}{557}a^{6}-\frac{3468}{557}a^{5}-\frac{2781}{557}a^{4}+\frac{7765}{557}a^{3}-\frac{6294}{557}a^{2}+\frac{2285}{557}a+\frac{394}{557}$, $\frac{514}{557}a^{11}-\frac{1670}{557}a^{10}+\frac{1175}{557}a^{9}+\frac{2782}{557}a^{8}-\frac{8323}{557}a^{7}+\frac{8656}{557}a^{6}+\frac{1713}{557}a^{5}-\frac{12441}{557}a^{4}+\frac{12265}{557}a^{3}-\frac{4586}{557}a^{2}-\frac{576}{557}a+\frac{797}{557}$, $\frac{129}{557}a^{11}-\frac{560}{557}a^{10}+\frac{931}{557}a^{9}+\frac{9}{557}a^{8}-\frac{2881}{557}a^{7}+\frac{5224}{557}a^{6}-\frac{3468}{557}a^{5}-\frac{2781}{557}a^{4}+\frac{7765}{557}a^{3}-\frac{5737}{557}a^{2}+\frac{1728}{557}a+\frac{394}{557}$ 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:  \( 13.1676614702 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 13.1676614702 \cdot 1}{2\cdot\sqrt{972292630401}}\cr\approx \mathstrut & 0.166502061927 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 5*x^10 + 3*x^9 - 20*x^8 + 30*x^7 - 12*x^6 - 24*x^5 + 43*x^4 - 30*x^3 + 8*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 - 4*x^11 + 5*x^10 + 3*x^9 - 20*x^8 + 30*x^7 - 12*x^6 - 24*x^5 + 43*x^4 - 30*x^3 + 8*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 - 4*x^11 + 5*x^10 + 3*x^9 - 20*x^8 + 30*x^7 - 12*x^6 - 24*x^5 + 43*x^4 - 30*x^3 + 8*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 - 4*x^11 + 5*x^10 + 3*x^9 - 20*x^8 + 30*x^7 - 12*x^6 - 24*x^5 + 43*x^4 - 30*x^3 + 8*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

$C_4^2:S_3$ (as 12T62):

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 96
The 10 conjugacy class representatives for $C_4^2:S_3$
Character table for $C_4^2:S_3$

Intermediate fields

3.1.331.1, 6.2.109561.1

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 sibling: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: 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 ${\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.4.0.1}{4} }$ R ${\href{/padicField/5.3.0.1}{3} }^{4}$ ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.3.0.1}{3} }^{4}$ ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ ${\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ ${\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.3.0.1}{3} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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
\(3\) Copy content Toggle raw display 3.4.0.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.2$x^{8} - 6 x^{6} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
\(331\) Copy content Toggle raw display $\Q_{331}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{331}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{331}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{331}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$