Properties

Label 12.0.49519263525896192.12
Degree $12$
Signature $[0, 6]$
Discriminant $4.952\times 10^{16}$
Root discriminant \(24.62\)
Ramified primes $2,7$
Class number $9$
Class group [3, 3]
Galois group $(C_3\times C_3):C_4$ (as 12T17)

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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 - 2*x^10 + 20*x^9 + 15*x^8 - 40*x^7 - 60*x^6 + 80*x^5 + 270*x^4 + 288*x^3 + 192*x^2 + 128*x + 64)
 
gp: K = bnfinit(y^12 - 4*y^11 - 2*y^10 + 20*y^9 + 15*y^8 - 40*y^7 - 60*y^6 + 80*y^5 + 270*y^4 + 288*y^3 + 192*y^2 + 128*y + 64, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - 2*x^10 + 20*x^9 + 15*x^8 - 40*x^7 - 60*x^6 + 80*x^5 + 270*x^4 + 288*x^3 + 192*x^2 + 128*x + 64);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 - 2*x^10 + 20*x^9 + 15*x^8 - 40*x^7 - 60*x^6 + 80*x^5 + 270*x^4 + 288*x^3 + 192*x^2 + 128*x + 64)
 

\( x^{12} - 4 x^{11} - 2 x^{10} + 20 x^{9} + 15 x^{8} - 40 x^{7} - 60 x^{6} + 80 x^{5} + 270 x^{4} + 288 x^{3} + 192 x^{2} + 128 x + 64 \) 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:   \(49519263525896192\) \(\medspace = 2^{33}\cdot 7^{8}\) 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:  \(24.62\)
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:  $2^{11/4}7^{2/3}\approx 24.616776431006123$
Ramified primes:   \(2\), \(7\) 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:  \(\Q(\sqrt{2}) \)
$\card{ \Aut(K/\Q) }$:  $6$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{9}+\frac{1}{4}a^{5}-\frac{1}{2}a$, $\frac{1}{8}a^{10}-\frac{1}{4}a^{7}+\frac{1}{8}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{90700304}a^{11}-\frac{915961}{45350152}a^{10}-\frac{1518341}{45350152}a^{9}+\frac{1998947}{22675076}a^{8}-\frac{5041965}{90700304}a^{7}-\frac{10847043}{45350152}a^{6}-\frac{345697}{11337538}a^{5}-\frac{1820374}{5668769}a^{4}+\frac{5236523}{45350152}a^{3}+\frac{418721}{1333828}a^{2}+\frac{5139753}{11337538}a+\frac{1929770}{5668769}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{3}\times C_{3}$, which has order $9$

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:   $\frac{55457}{2667656}a^{11}-\frac{221449}{2667656}a^{10}-\frac{75939}{1333828}a^{9}+\frac{650113}{1333828}a^{8}+\frac{778291}{2667656}a^{7}-\frac{3081319}{2667656}a^{6}-\frac{750827}{666914}a^{5}+\frac{1501473}{666914}a^{4}+\frac{7492991}{1333828}a^{3}+\frac{6558133}{1333828}a^{2}+\frac{573462}{333457}a+\frac{130991}{333457}$, $\frac{1484913}{90700304}a^{11}-\frac{3312685}{45350152}a^{10}+\frac{1192423}{45350152}a^{9}+\frac{4237907}{22675076}a^{8}+\frac{22569787}{90700304}a^{7}-\frac{15051799}{45350152}a^{6}-\frac{11598873}{11337538}a^{5}+\frac{4461267}{5668769}a^{4}+\frac{180066035}{45350152}a^{3}+\frac{8337041}{1333828}a^{2}+\frac{52278719}{11337538}a+\frac{6174355}{5668769}$, $\frac{891139}{90700304}a^{11}-\frac{2521269}{45350152}a^{10}+\frac{2917135}{45350152}a^{9}+\frac{666595}{5668769}a^{8}-\frac{2663659}{90700304}a^{7}-\frac{19553785}{45350152}a^{6}-\frac{496347}{11337538}a^{5}+\frac{11036105}{11337538}a^{4}+\frac{52604277}{45350152}a^{3}+\frac{230219}{1333828}a^{2}-\frac{6290415}{11337538}a-\frac{1462117}{5668769}$, $\frac{245257}{45350152}a^{11}-\frac{2069321}{45350152}a^{10}+\frac{1672871}{11337538}a^{9}-\frac{2073817}{11337538}a^{8}-\frac{3277883}{45350152}a^{7}+\frac{17776839}{45350152}a^{6}+\frac{668839}{11337538}a^{5}-\frac{2783201}{5668769}a^{4}-\frac{11045691}{22675076}a^{3}+\frac{2009127}{1333828}a^{2}+\frac{13245298}{5668769}a-\frac{8852147}{5668769}$, $\frac{1670881}{45350152}a^{11}-\frac{9485273}{45350152}a^{10}+\frac{2702997}{11337538}a^{9}+\frac{12766549}{22675076}a^{8}-\frac{29859809}{45350152}a^{7}-\frac{46685543}{45350152}a^{6}+\frac{8341561}{22675076}a^{5}+\frac{42466891}{11337538}a^{4}+\frac{65910947}{22675076}a^{3}+\frac{1325093}{1333828}a^{2}+\frac{22402917}{11337538}a+\frac{758957}{5668769}$ 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:  \( 2643.94789139 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 2643.94789139 \cdot 9}{2\cdot\sqrt{49519263525896192}}\cr\approx \mathstrut & 3.28970933319 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 2*x^10 + 20*x^9 + 15*x^8 - 40*x^7 - 60*x^6 + 80*x^5 + 270*x^4 + 288*x^3 + 192*x^2 + 128*x + 64)
 
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 - 2*x^10 + 20*x^9 + 15*x^8 - 40*x^7 - 60*x^6 + 80*x^5 + 270*x^4 + 288*x^3 + 192*x^2 + 128*x + 64, 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 - 2*x^10 + 20*x^9 + 15*x^8 - 40*x^7 - 60*x^6 + 80*x^5 + 270*x^4 + 288*x^3 + 192*x^2 + 128*x + 64);
 
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 - 2*x^10 + 20*x^9 + 15*x^8 - 40*x^7 - 60*x^6 + 80*x^5 + 270*x^4 + 288*x^3 + 192*x^2 + 128*x + 64);
 
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

$C_3^2:C_4$ (as 12T17):

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 36
The 6 conjugacy class representatives for $(C_3\times C_3):C_4$
Character table for $(C_3\times C_3):C_4$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.2048.2, 6.2.39337984.1 x3

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 36
Degree 6 siblings: 6.2.802816.1, 6.2.39337984.1
Degree 9 sibling: 9.1.493455671296.1
Degree 12 sibling: 12.0.20624432955392.2
Degree 18 sibling: 18.2.1947987996273487986556928.1
Minimal sibling: 6.2.802816.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{3}$ ${\href{/padicField/5.4.0.1}{4} }^{3}$ R ${\href{/padicField/11.4.0.1}{4} }^{3}$ ${\href{/padicField/13.4.0.1}{4} }^{3}$ ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{6}$ ${\href{/padicField/19.4.0.1}{4} }^{3}$ ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{6}$ ${\href{/padicField/29.4.0.1}{4} }^{3}$ ${\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.2.0.1}{2} }^{6}$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.4.0.1}{4} }^{3}$

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 2.4.11.2$x^{4} + 4 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 4 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 4 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
\(7\) Copy content Toggle raw display 7.3.2.3$x^{3} + 21$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} + 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} + 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} + 21$$3$$1$$2$$C_3$$[\ ]_{3}$