Properties

Label 12.4.8067775586689.3
Degree $12$
Signature $[4, 4]$
Discriminant $8.068\times 10^{12}$
Root discriminant \(11.90\)
Ramified primes $7,13$
Class number $1$
Class group trivial
Galois group $A_4\times C_2$ (as 12T6)

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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 + 2*x^10 + x^9 + 4*x^8 - 5*x^7 - 20*x^6 + 28*x^5 - 21*x^4 + 35*x^3 - 21*x^2 + 7*x - 7)
 
gp: K = bnfinit(y^12 - 3*y^11 + 2*y^10 + y^9 + 4*y^8 - 5*y^7 - 20*y^6 + 28*y^5 - 21*y^4 + 35*y^3 - 21*y^2 + 7*y - 7, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 + 2*x^10 + x^9 + 4*x^8 - 5*x^7 - 20*x^6 + 28*x^5 - 21*x^4 + 35*x^3 - 21*x^2 + 7*x - 7);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 3*x^11 + 2*x^10 + x^9 + 4*x^8 - 5*x^7 - 20*x^6 + 28*x^5 - 21*x^4 + 35*x^3 - 21*x^2 + 7*x - 7)
 

\( x^{12} - 3x^{11} + 2x^{10} + x^{9} + 4x^{8} - 5x^{7} - 20x^{6} + 28x^{5} - 21x^{4} + 35x^{3} - 21x^{2} + 7x - 7 \) 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:   \(8067775586689\) \(\medspace = 7^{10}\cdot 13^{4}\) 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:  \(11.90\)
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:  $7^{5/6}13^{1/2}\approx 18.248200448594663$
Ramified primes:   \(7\), \(13\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5}a^{7}-\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{2}{5}a^{3}-\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{8}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{2}{5}a^{4}+\frac{2}{5}a^{3}-\frac{2}{5}a^{2}+\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{9}-\frac{2}{5}a^{6}+\frac{2}{5}a^{4}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{10}-\frac{2}{5}a^{6}+\frac{1}{5}a^{5}+\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{8425}a^{11}-\frac{354}{8425}a^{10}-\frac{434}{8425}a^{9}+\frac{137}{1685}a^{8}+\frac{524}{8425}a^{7}-\frac{3634}{8425}a^{6}+\frac{1654}{8425}a^{5}+\frac{2484}{8425}a^{4}-\frac{152}{1685}a^{3}-\frac{224}{1685}a^{2}-\frac{2876}{8425}a-\frac{3202}{8425}$ 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:   $\frac{23}{8425}a^{11}+\frac{283}{8425}a^{10}-\frac{1557}{8425}a^{9}+\frac{91}{337}a^{8}+\frac{257}{8425}a^{7}-\frac{1017}{8425}a^{6}-\frac{4083}{8425}a^{5}-\frac{3528}{8425}a^{4}+\frac{4929}{1685}a^{3}-\frac{963}{337}a^{2}+\frac{6307}{8425}a-\frac{6246}{8425}$, $\frac{1228}{8425}a^{11}-\frac{1667}{8425}a^{10}-\frac{2177}{8425}a^{9}+\frac{82}{337}a^{8}+\frac{8227}{8425}a^{7}+\frac{2698}{8425}a^{6}-\frac{29643}{8425}a^{5}-\frac{6238}{8425}a^{4}+\frac{1053}{1685}a^{3}+\frac{3964}{1685}a^{2}+\frac{10142}{8425}a-\frac{4321}{8425}$, $\frac{1228}{8425}a^{11}-\frac{1667}{8425}a^{10}-\frac{2177}{8425}a^{9}+\frac{82}{337}a^{8}+\frac{8227}{8425}a^{7}+\frac{2698}{8425}a^{6}-\frac{29643}{8425}a^{5}-\frac{6238}{8425}a^{4}+\frac{1053}{1685}a^{3}+\frac{3964}{1685}a^{2}+\frac{10142}{8425}a+\frac{4104}{8425}$, $\frac{134}{1685}a^{11}-\frac{593}{1685}a^{10}+\frac{819}{1685}a^{9}-\frac{211}{1685}a^{8}+\frac{24}{337}a^{7}-\frac{1002}{1685}a^{6}-\frac{359}{337}a^{5}+\frac{6977}{1685}a^{4}-\frac{8491}{1685}a^{3}+\frac{1662}{337}a^{2}-\frac{5248}{1685}a+\frac{1618}{1685}$, $\frac{878}{8425}a^{11}-\frac{2457}{8425}a^{10}-\frac{242}{8425}a^{9}+\frac{988}{1685}a^{8}+\frac{3437}{8425}a^{7}-\frac{7687}{8425}a^{6}-\frac{25533}{8425}a^{5}+\frac{27522}{8425}a^{4}+\frac{4377}{1685}a^{3}-\frac{201}{1685}a^{2}-\frac{12793}{8425}a-\frac{22681}{8425}$, $\frac{638}{1685}a^{11}-\frac{282}{337}a^{10}+\frac{459}{1685}a^{9}+\frac{278}{1685}a^{8}+\frac{3041}{1685}a^{7}-\frac{269}{1685}a^{6}-\frac{12027}{1685}a^{5}+\frac{7632}{1685}a^{4}-\frac{12743}{1685}a^{3}+\frac{18078}{1685}a^{2}-\frac{726}{337}a+\frac{6084}{1685}$, $\frac{72}{1685}a^{11}+\frac{124}{1685}a^{10}-\frac{581}{1685}a^{9}+\frac{118}{1685}a^{8}+\frac{199}{337}a^{7}+\frac{175}{337}a^{6}-\frac{379}{337}a^{5}-\frac{896}{337}a^{4}+\frac{3244}{1685}a^{3}-\frac{289}{337}a^{2}+\frac{1531}{1685}a+\frac{1649}{1685}$ 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:  \( 51.9911514065 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 51.9911514065 \cdot 1}{2\cdot\sqrt{8067775586689}}\cr\approx \mathstrut & 0.228224356518 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 2*x^10 + x^9 + 4*x^8 - 5*x^7 - 20*x^6 + 28*x^5 - 21*x^4 + 35*x^3 - 21*x^2 + 7*x - 7)
 
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 + 2*x^10 + x^9 + 4*x^8 - 5*x^7 - 20*x^6 + 28*x^5 - 21*x^4 + 35*x^3 - 21*x^2 + 7*x - 7, 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 + 2*x^10 + x^9 + 4*x^8 - 5*x^7 - 20*x^6 + 28*x^5 - 21*x^4 + 35*x^3 - 21*x^2 + 7*x - 7);
 
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 + 2*x^10 + x^9 + 4*x^8 - 5*x^7 - 20*x^6 + 28*x^5 - 21*x^4 + 35*x^3 - 21*x^2 + 7*x - 7);
 
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

$C_2\times A_4$ (as 12T6):

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 8 conjugacy class representatives for $A_4\times C_2$
Character table for $A_4\times C_2$

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.4.218491.1 x2, 6.2.405769.1

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 6 sibling: 6.4.218491.1
Degree 8 sibling: 8.0.3360173089.1
Degree 12 sibling: 12.0.1363454074150441.3
Minimal sibling: 6.4.218491.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }^{2}$ ${\href{/padicField/3.6.0.1}{6} }^{2}$ ${\href{/padicField/5.3.0.1}{3} }^{4}$ R ${\href{/padicField/11.6.0.1}{6} }^{2}$ R ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.3.0.1}{3} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{6}$ ${\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.6.0.1}{6} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.2.0.1}{2} }^{6}$ ${\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.3.0.1}{3} }^{4}$ ${\href{/padicField/59.3.0.1}{3} }^{4}$

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
\(7\) Copy content Toggle raw display 7.6.5.2$x^{6} + 42$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} + 42$$6$$1$$5$$C_6$$[\ ]_{6}$
\(13\) Copy content Toggle raw display 13.2.1.1$x^{2} + 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} + 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$