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Properties

Label 12.0.41085390865563648.20
Degree $12$
Signature $[0, 6]$
Discriminant $4.109\times 10^{16}$
Root discriminant \(24.24\)
Ramified primes $2,3$
Class number $2$
Class group [2]
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 + 12*x^10 - 8*x^9 + 78*x^8 - 72*x^7 + 308*x^6 - 288*x^5 + 711*x^4 - 592*x^3 + 924*x^2 - 816*x + 526)
 
gp: K = bnfinit(y^12 + 12*y^10 - 8*y^9 + 78*y^8 - 72*y^7 + 308*y^6 - 288*y^5 + 711*y^4 - 592*y^3 + 924*y^2 - 816*y + 526, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 12*x^10 - 8*x^9 + 78*x^8 - 72*x^7 + 308*x^6 - 288*x^5 + 711*x^4 - 592*x^3 + 924*x^2 - 816*x + 526);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 + 12*x^10 - 8*x^9 + 78*x^8 - 72*x^7 + 308*x^6 - 288*x^5 + 711*x^4 - 592*x^3 + 924*x^2 - 816*x + 526)
 

\( x^{12} + 12 x^{10} - 8 x^{9} + 78 x^{8} - 72 x^{7} + 308 x^{6} - 288 x^{5} + 711 x^{4} - 592 x^{3} + \cdots + 526 \) 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:   \(41085390865563648\) \(\medspace = 2^{33}\cdot 3^{14}\) 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:  \(24.24\)
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:  $2^{11/4}3^{25/18}\approx 30.93861708477606$
Ramified primes:   \(2\), \(3\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}$, $\frac{1}{92}a^{10}-\frac{3}{23}a^{9}-\frac{9}{92}a^{8}-\frac{9}{23}a^{7}-\frac{29}{92}a^{6}-\frac{10}{23}a^{5}-\frac{39}{92}a^{4}-\frac{7}{23}a^{3}+\frac{15}{46}a^{2}-\frac{3}{23}a-\frac{9}{46}$, $\frac{1}{100147246024}a^{11}+\frac{433626103}{100147246024}a^{10}+\frac{17866447045}{100147246024}a^{9}+\frac{11609465747}{100147246024}a^{8}+\frac{33759513395}{100147246024}a^{7}-\frac{3161873683}{100147246024}a^{6}+\frac{20217800271}{100147246024}a^{5}+\frac{9691153321}{100147246024}a^{4}-\frac{3572510263}{7153374716}a^{3}+\frac{8483199065}{50073623012}a^{2}+\frac{5707193217}{50073623012}a-\frac{10981382801}{50073623012}$ 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

$C_{2}$, which has order $2$

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:   $\frac{15519}{9147538}a^{11}-\frac{18711}{18295076}a^{10}+\frac{104444}{4573769}a^{9}-\frac{370185}{18295076}a^{8}+\frac{1521663}{9147538}a^{7}-\frac{3660897}{18295076}a^{6}+\frac{2920773}{4573769}a^{5}-\frac{15852027}{18295076}a^{4}+\frac{6738645}{4573769}a^{3}-\frac{16606359}{9147538}a^{2}+\frac{6275490}{4573769}a-\frac{7449199}{9147538}$, $\frac{1404788337}{50073623012}a^{11}+\frac{965501839}{50073623012}a^{10}+\frac{17049697447}{50073623012}a^{9}+\frac{208775229}{50073623012}a^{8}+\frac{104071766763}{50073623012}a^{7}-\frac{30246176579}{50073623012}a^{6}+\frac{377081683073}{50073623012}a^{5}-\frac{141667545877}{50073623012}a^{4}+\frac{55581814627}{3576687358}a^{3}-\frac{145750375675}{25036811506}a^{2}+\frac{414792347485}{25036811506}a-\frac{287258155965}{25036811506}$, $\frac{68285995}{100147246024}a^{11}+\frac{330117327}{100147246024}a^{10}+\frac{1127310323}{100147246024}a^{9}+\frac{2246928255}{100147246024}a^{8}+\frac{5678106377}{100147246024}a^{7}+\frac{8010342925}{100147246024}a^{6}+\frac{9858381201}{100147246024}a^{5}+\frac{27053130653}{100147246024}a^{4}-\frac{773861201}{7153374716}a^{3}+\frac{21977231029}{50073623012}a^{2}-\frac{18472337305}{50073623012}a+\frac{15494938903}{50073623012}$, $\frac{96126309}{14306749432}a^{11}+\frac{102315549}{14306749432}a^{10}+\frac{1088929833}{14306749432}a^{9}+\frac{696868177}{14306749432}a^{8}+\frac{6246801071}{14306749432}a^{7}+\frac{1941946199}{14306749432}a^{6}+\frac{16368494667}{14306749432}a^{5}+\frac{3544258147}{14306749432}a^{4}+\frac{10919660759}{7153374716}a^{3}-\frac{1106500961}{7153374716}a^{2}+\frac{5377155909}{7153374716}a+\frac{202027135}{311016292}$, $\frac{78345697}{25036811506}a^{11}-\frac{178004486}{12518405753}a^{10}+\frac{408338926}{12518405753}a^{9}-\frac{4253028401}{25036811506}a^{8}+\frac{8039388669}{25036811506}a^{7}-\frac{13323635676}{12518405753}a^{6}+\frac{19334803844}{12518405753}a^{5}-\frac{90321137107}{25036811506}a^{4}+\frac{7398587056}{1788343679}a^{3}-\frac{79995512155}{12518405753}a^{2}+\frac{49557442635}{12518405753}a-\frac{22674771670}{12518405753}$ 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:  \( 6058.14180821 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 6058.14180821 \cdot 2}{2\cdot\sqrt{41085390865563648}}\cr\approx \mathstrut & 1.83897122475 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 + 12*x^10 - 8*x^9 + 78*x^8 - 72*x^7 + 308*x^6 - 288*x^5 + 711*x^4 - 592*x^3 + 924*x^2 - 816*x + 526)
 
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 + 12*x^10 - 8*x^9 + 78*x^8 - 72*x^7 + 308*x^6 - 288*x^5 + 711*x^4 - 592*x^3 + 924*x^2 - 816*x + 526, 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 + 12*x^10 - 8*x^9 + 78*x^8 - 72*x^7 + 308*x^6 - 288*x^5 + 711*x^4 - 592*x^3 + 924*x^2 - 816*x + 526);
 
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 + 12*x^10 - 8*x^9 + 78*x^8 - 72*x^7 + 308*x^6 - 288*x^5 + 711*x^4 - 592*x^3 + 924*x^2 - 816*x + 526);
 
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

\(\Q(\sqrt{2}) \), 4.0.18432.2, 6.2.11943936.2 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: 6.2.11943936.1, 6.2.11943936.2
Degree 9 sibling: 9.1.2229025112064.1
Degree 12 sibling: 12.0.41085390865563648.12
Degree 18 sibling: deg 18
Minimal sibling: 6.2.11943936.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.4.0.1}{4} }^{3}$ ${\href{/padicField/7.2.0.1}{2} }^{6}$ ${\href{/padicField/11.4.0.1}{4} }^{3}$ ${\href{/padicField/13.4.0.1}{4} }^{3}$ ${\href{/padicField/17.2.0.1}{2} }^{6}$ ${\href{/padicField/19.4.0.1}{4} }^{3}$ ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{6}$ ${\href{/padicField/29.4.0.1}{4} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.3.0.1}{3} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.2.0.1}{2} }^{6}$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.4.0.1}{4} }^{3}$

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 2.4.11.4$x^{4} + 8 x^{3} + 4 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.4$x^{4} + 8 x^{3} + 4 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.4$x^{4} + 8 x^{3} + 4 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
\(3\) Copy content Toggle raw display 3.12.14.1$x^{12} - 6 x^{8} - 6 x^{6} + 18 x^{4} + 36 x^{2} + 18$$6$$2$$14$$(C_3\times C_3):C_4$$[3/2, 3/2]_{2}^{2}$