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Properties

Label 12.6.623334901029867.1
Degree $12$
Signature $[6, 3]$
Discriminant $-6.233\times 10^{14}$
Root discriminant \(17.10\)
Ramified primes $3,13$
Class number $1$
Class group trivial
Galois group $D_4 \times C_3$ (as 12T14)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^10 - 8*x^9 - 9*x^8 + 21*x^7 - 2*x^6 - 27*x^5 + 12*x^4 + 45*x^3 + 72*x^2 + 60*x + 17)
 
gp: K = bnfinit(y^12 - 3*y^10 - 8*y^9 - 9*y^8 + 21*y^7 - 2*y^6 - 27*y^5 + 12*y^4 + 45*y^3 + 72*y^2 + 60*y + 17, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^10 - 8*x^9 - 9*x^8 + 21*x^7 - 2*x^6 - 27*x^5 + 12*x^4 + 45*x^3 + 72*x^2 + 60*x + 17);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 3*x^10 - 8*x^9 - 9*x^8 + 21*x^7 - 2*x^6 - 27*x^5 + 12*x^4 + 45*x^3 + 72*x^2 + 60*x + 17)
 

\( x^{12} - 3x^{10} - 8x^{9} - 9x^{8} + 21x^{7} - 2x^{6} - 27x^{5} + 12x^{4} + 45x^{3} + 72x^{2} + 60x + 17 \) 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:  $[6, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-623334901029867\) \(\medspace = -\,3^{17}\cdot 13^{6}\) 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:  \(17.10\)
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:  $3^{3/2}13^{1/2}\approx 18.734993995195193$
Ramified primes:   \(3\), \(13\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $6$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{70179281}a^{11}+\frac{23903892}{70179281}a^{10}-\frac{3831727}{70179281}a^{9}-\frac{1952393}{4128193}a^{8}+\frac{2415317}{70179281}a^{7}+\frac{34924300}{70179281}a^{6}+\frac{5112849}{70179281}a^{5}-\frac{17015376}{70179281}a^{4}-\frac{2467102}{70179281}a^{3}-\frac{5633895}{70179281}a^{2}-\frac{12579417}{70179281}a-\frac{26398}{4128193}$ 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:  $8$
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:   $\frac{278734842}{70179281}a^{11}-\frac{186820278}{70179281}a^{10}-\frac{712093707}{70179281}a^{9}-\frac{103026078}{4128193}a^{8}-\frac{1331732070}{70179281}a^{7}+\frac{6751465727}{70179281}a^{6}-\frac{5080225473}{70179281}a^{5}-\frac{4152326643}{70179281}a^{4}+\frac{6154287726}{70179281}a^{3}+\frac{8410280175}{70179281}a^{2}+\frac{14376875841}{70179281}a+\frac{416253261}{4128193}$, $\frac{278734842}{70179281}a^{11}-\frac{186820278}{70179281}a^{10}-\frac{712093707}{70179281}a^{9}-\frac{103026078}{4128193}a^{8}-\frac{1331732070}{70179281}a^{7}+\frac{6751465727}{70179281}a^{6}-\frac{5080225473}{70179281}a^{5}-\frac{4152326643}{70179281}a^{4}+\frac{6154287726}{70179281}a^{3}+\frac{8410280175}{70179281}a^{2}+\frac{14376875841}{70179281}a+\frac{420381454}{4128193}$, $\frac{47128729}{4128193}a^{11}-\frac{30810473}{4128193}a^{10}-\frac{121307647}{4128193}a^{9}-\frac{297656554}{4128193}a^{8}-\frac{229387873}{4128193}a^{7}+\frac{1140114694}{4128193}a^{6}-\frac{839444589}{4128193}a^{5}-\frac{725836865}{4128193}a^{4}+\frac{1040214878}{4128193}a^{3}+\frac{1440491679}{4128193}a^{2}+\frac{2452509623}{4128193}a+\frac{1223790350}{4128193}$, $\frac{16494206}{4128193}a^{11}-\frac{10654986}{4128193}a^{10}-\frac{42588417}{4128193}a^{9}-\frac{104518682}{4128193}a^{8}-\frac{80778723}{4128193}a^{7}+\frac{398770852}{4128193}a^{6}-\frac{290458202}{4128193}a^{5}-\frac{258446475}{4128193}a^{4}+\frac{361067188}{4128193}a^{3}+\frac{510636268}{4128193}a^{2}+\frac{859729384}{4128193}a+\frac{430428154}{4128193}$, $\frac{50679932}{70179281}a^{11}-\frac{40514360}{70179281}a^{10}-\frac{119291646}{70179281}a^{9}-\frac{18019879}{4128193}a^{8}-\frac{214725219}{70179281}a^{7}+\frac{1224081612}{70179281}a^{6}-\frac{1101880949}{70179281}a^{5}-\frac{498606154}{70179281}a^{4}+\frac{1091169528}{70179281}a^{3}+\frac{1305169633}{70179281}a^{2}+\frac{2486161011}{70179281}a+\frac{69609013}{4128193}$, $\frac{306906874}{70179281}a^{11}-\frac{198004021}{70179281}a^{10}-\frac{793635044}{70179281}a^{9}-\frac{114101659}{4128193}a^{8}-\frac{1511548468}{70179281}a^{7}+\frac{7414640173}{70179281}a^{6}-\frac{5410412598}{70179281}a^{5}-\frac{4817074291}{70179281}a^{4}+\frac{6893754680}{70179281}a^{3}+\frac{9286267879}{70179281}a^{2}+\frac{16006784429}{70179281}a+\frac{482716177}{4128193}$, $\frac{520743348}{70179281}a^{11}-\frac{360896124}{70179281}a^{10}-\frac{1309489086}{70179281}a^{9}-\frac{191967922}{4128193}a^{8}-\frac{2427582181}{70179281}a^{7}+\frac{12608890403}{70179281}a^{6}-\frac{9773948940}{70179281}a^{5}-\frac{7225996169}{70179281}a^{4}+\frac{11104978219}{70179281}a^{3}+\frac{15777465596}{70179281}a^{2}+\frac{26652454603}{70179281}a+\frac{758018919}{4128193}$, $\frac{278987432}{70179281}a^{11}-\frac{177180833}{70179281}a^{10}-\frac{725552366}{70179281}a^{9}-\frac{104038168}{4128193}a^{8}-\frac{1385480054}{70179281}a^{7}+\frac{6744781027}{70179281}a^{6}-\frac{4854287682}{70179281}a^{5}-\frac{4497519886}{70179281}a^{4}+\frac{6251188107}{70179281}a^{3}+\frac{8580560805}{70179281}a^{2}+\frac{14659779491}{70179281}a+\frac{436055101}{4128193}$ 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:  \( 1028.79993348 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{3}\cdot 1028.79993348 \cdot 1}{2\cdot\sqrt{623334901029867}}\cr\approx \mathstrut & 0.327084368012 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^10 - 8*x^9 - 9*x^8 + 21*x^7 - 2*x^6 - 27*x^5 + 12*x^4 + 45*x^3 + 72*x^2 + 60*x + 17)
 
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^10 - 8*x^9 - 9*x^8 + 21*x^7 - 2*x^6 - 27*x^5 + 12*x^4 + 45*x^3 + 72*x^2 + 60*x + 17, 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^10 - 8*x^9 - 9*x^8 + 21*x^7 - 2*x^6 - 27*x^5 + 12*x^4 + 45*x^3 + 72*x^2 + 60*x + 17);
 
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^10 - 8*x^9 - 9*x^8 + 21*x^7 - 2*x^6 - 27*x^5 + 12*x^4 + 45*x^3 + 72*x^2 + 60*x + 17);
 
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

$C_3\times D_4$ (as 12T14):

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 15 conjugacy class representatives for $D_4 \times C_3$
Character table for $D_4 \times C_3$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{9})^+\), 4.2.507.1, 6.6.14414517.1

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 12 sibling: 12.0.851162814333.2
Minimal sibling: 12.0.851162814333.2

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 ${\href{/padicField/5.12.0.1}{12} }$ ${\href{/padicField/7.6.0.1}{6} }^{2}$ ${\href{/padicField/11.12.0.1}{12} }$ R ${\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{6}$ ${\href{/padicField/19.2.0.1}{2} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{6}$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{6}$ ${\href{/padicField/59.12.0.1}{12} }$

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 3.6.8.3$x^{6} + 18 x^{5} + 114 x^{4} + 326 x^{3} + 570 x^{2} + 528 x + 197$$3$$2$$8$$C_6$$[2]^{2}$
3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[2]_{2}$
\(13\) Copy content Toggle raw display 13.12.6.1$x^{12} + 780 x^{11} + 253578 x^{10} + 43990720 x^{9} + 4297346257 x^{8} + 224493831662 x^{7} + 4938918346310 x^{6} + 2961720498866 x^{5} + 3005850529646 x^{4} + 51307643736852 x^{3} + 70292613843513 x^{2} + 65587287977710 x + 15475747398037$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$