Properties

Label 12.0.2658691056664576.10
Degree $12$
Signature $[0, 6]$
Discriminant $2.659\times 10^{15}$
Root discriminant \(19.29\)
Ramified primes $2,19$
Class number $1$
Class group trivial
Galois group $A_4^2:D_4$ (as 12T208)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 2*x^10 + 7*x^8 - 2*x^6 + 19)
 
gp: K = bnfinit(y^12 + 2*y^10 + 7*y^8 - 2*y^6 + 19, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 2*x^10 + 7*x^8 - 2*x^6 + 19);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 + 2*x^10 + 7*x^8 - 2*x^6 + 19)
 

\( x^{12} + 2x^{10} + 7x^{8} - 2x^{6} + 19 \) 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:   \(2658691056664576\) \(\medspace = 2^{30}\cdot 19^{5}\) 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:  \(19.29\)
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:  $2^{215/64}19^{5/6}\approx 119.37149761252084$
Ramified primes:   \(2\), \(19\) 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:  \(\Q(\sqrt{19}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2247}a^{10}-\frac{2}{321}a^{8}+\frac{11}{107}a^{6}+\frac{796}{2247}a^{4}+\frac{746}{2247}a^{2}-\frac{701}{2247}$, $\frac{1}{2247}a^{11}-\frac{2}{321}a^{9}+\frac{11}{107}a^{7}+\frac{796}{2247}a^{5}+\frac{746}{2247}a^{3}-\frac{701}{2247}a$ 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:   $\frac{131}{2247}a^{10}+\frac{59}{321}a^{8}+\frac{50}{107}a^{6}+\frac{914}{2247}a^{4}-\frac{1142}{2247}a^{2}+\frac{2543}{2247}$, $\frac{17}{2247}a^{10}-\frac{34}{321}a^{8}-\frac{27}{107}a^{6}-\frac{2197}{2247}a^{4}-\frac{800}{2247}a^{2}-\frac{682}{2247}$, $\frac{148}{2247}a^{11}-\frac{53}{749}a^{10}+\frac{25}{321}a^{9}-\frac{1}{107}a^{8}+\frac{23}{107}a^{7}-\frac{37}{107}a^{6}-\frac{1283}{2247}a^{5}+\frac{505}{749}a^{4}-\frac{1942}{2247}a^{3}-\frac{590}{749}a^{2}+\frac{1861}{2247}a-\frac{297}{749}$, $\frac{47}{749}a^{11}+\frac{8}{2247}a^{10}+\frac{13}{107}a^{9}-\frac{16}{321}a^{8}+\frac{53}{107}a^{7}-\frac{19}{107}a^{6}-\frac{38}{749}a^{5}-\frac{373}{2247}a^{4}-\frac{141}{749}a^{3}+\frac{1474}{2247}a^{2}-\frac{740}{749}a+\frac{3380}{2247}$, $\frac{121}{2247}a^{11}+\frac{13}{2247}a^{10}+\frac{79}{321}a^{9}-\frac{26}{321}a^{8}+\frac{47}{107}a^{7}+\frac{36}{107}a^{6}-\frac{305}{2247}a^{5}-\frac{887}{2247}a^{4}-\frac{1861}{2247}a^{3}-\frac{1537}{2247}a^{2}-\frac{1682}{2247}a-\frac{4619}{2247}$ 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:  \( 1021.08997934 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1021.08997934 \cdot 1}{2\cdot\sqrt{2658691056664576}}\cr\approx \mathstrut & 0.609227208980 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 + 2*x^10 + 7*x^8 - 2*x^6 + 19)
 
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 + 2*x^10 + 7*x^8 - 2*x^6 + 19, 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 + 2*x^10 + 7*x^8 - 2*x^6 + 19);
 
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 + 2*x^10 + 7*x^8 - 2*x^6 + 19);
 
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

$A_4^2:D_4$ (as 12T208):

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 1152
The 44 conjugacy class representatives for $A_4^2:D_4$
Character table for $A_4^2:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-2}) \), 6.0.184832.1

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 sibling: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: 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 ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ ${\href{/padicField/5.6.0.1}{6} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.12.0.1}{12} }$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ R ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.12.0.1}{12} }$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.12.0.1}{12} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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 2.12.30.170$x^{12} + 128 x^{9} - 50 x^{8} - 1888 x^{7} - 2144 x^{6} - 1536 x^{5} - 548 x^{4} - 768 x^{3} - 1024 x^{2} + 712$$4$$3$$30$12T134$[2, 2, 2, 3, 7/2, 7/2, 7/2]^{3}$
\(19\) Copy content Toggle raw display 19.6.5.4$x^{6} + 76$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.0.1$x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$