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Properties

Label 12.0.313...352.1
Degree $12$
Signature $[0, 6]$
Discriminant $3.134\times 10^{27}$
Root discriminant \(195.59\)
Ramified primes $2,3,7,11$
Class number $1639840$ (GRH)
Class group [2, 2, 2, 204980] (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 924*x^10 + 289674*x^8 + 38236968*x^6 + 2324405160*x^4 + 61364296224*x^2 + 506255443848)
 
gp: K = bnfinit(y^12 + 924*y^10 + 289674*y^8 + 38236968*y^6 + 2324405160*y^4 + 61364296224*y^2 + 506255443848, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 924*x^10 + 289674*x^8 + 38236968*x^6 + 2324405160*x^4 + 61364296224*x^2 + 506255443848);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 + 924*x^10 + 289674*x^8 + 38236968*x^6 + 2324405160*x^4 + 61364296224*x^2 + 506255443848)
 

\( x^{12} + 924x^{10} + 289674x^{8} + 38236968x^{6} + 2324405160x^{4} + 61364296224x^{2} + 506255443848 \) 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:   \(3133674591916081765366628352\) \(\medspace = 2^{33}\cdot 3^{6}\cdot 7^{10}\cdot 11^{6}\) 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:  \(195.59\)
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:  $2^{11/4}3^{1/2}7^{5/6}11^{1/2}\approx 195.58602694305185$
Ramified primes:   \(2\), \(3\), \(7\), \(11\) 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:  \(\Q(\sqrt{2}) \)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3696=2^{4}\cdot 3\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3696}(1,·)$, $\chi_{3696}(2309,·)$, $\chi_{3696}(529,·)$, $\chi_{3696}(2377,·)$, $\chi_{3696}(461,·)$, $\chi_{3696}(3629,·)$, $\chi_{3696}(2641,·)$, $\chi_{3696}(1781,·)$, $\chi_{3696}(793,·)$, $\chi_{3696}(1849,·)$, $\chi_{3696}(1517,·)$, $\chi_{3696}(3365,·)$$\rbrace$
This is a CM field.
Reflex fields:  4.0.109283328.8$^{2}$, 12.0.3133674591916081765366628352.1$^{30}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{33}a^{2}$, $\frac{1}{33}a^{3}$, $\frac{1}{2178}a^{4}$, $\frac{1}{2178}a^{5}$, $\frac{1}{503118}a^{6}$, $\frac{1}{503118}a^{7}$, $\frac{1}{33205788}a^{8}$, $\frac{1}{33205788}a^{9}$, $\frac{1}{7670537028}a^{10}+\frac{1}{5082}a^{4}$, $\frac{1}{7670537028}a^{11}+\frac{1}{5082}a^{5}$ 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

$C_{2}\times C_{2}\times C_{2}\times C_{204980}$, which has order $1639840$ (assuming GRH)

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:   $\frac{1}{547895502}a^{10}+\frac{13}{8301447}a^{8}+\frac{107}{251559}a^{6}+\frac{91}{2178}a^{4}+\frac{50}{33}a^{2}+14$, $\frac{1}{33205788}a^{8}+\frac{2}{83853}a^{6}+\frac{2}{363}a^{4}+\frac{4}{11}a^{2}+5$, $\frac{1}{697321548}a^{10}+\frac{10}{8301447}a^{8}+\frac{79}{251559}a^{6}+\frac{24}{847}a^{4}+\frac{31}{33}a^{2}+9$, $\frac{1}{697321548}a^{10}+\frac{13}{11068596}a^{8}+\frac{73}{251559}a^{6}+\frac{58}{2541}a^{4}+\frac{19}{33}a^{2}+4$, $\frac{25}{7670537028}a^{10}+\frac{23}{8301447}a^{8}+\frac{62}{83853}a^{6}+\frac{1069}{15246}a^{4}+\frac{27}{11}a^{2}+24$ Copy content Toggle raw display (assuming GRH)
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:  \( 279.1500271937239 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 279.1500271937239 \cdot 1639840}{2\cdot\sqrt{3133674591916081765366628352}}\cr\approx \mathstrut & 0.251571486875678 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 + 924*x^10 + 289674*x^8 + 38236968*x^6 + 2324405160*x^4 + 61364296224*x^2 + 506255443848)
 
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 + 924*x^10 + 289674*x^8 + 38236968*x^6 + 2324405160*x^4 + 61364296224*x^2 + 506255443848, 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 + 924*x^10 + 289674*x^8 + 38236968*x^6 + 2324405160*x^4 + 61364296224*x^2 + 506255443848);
 
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 + 924*x^10 + 289674*x^8 + 38236968*x^6 + 2324405160*x^4 + 61364296224*x^2 + 506255443848);
 
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

$C_{12}$ (as 12T1):

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 cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{7})^+\), 4.0.109283328.8, 6.6.1229312.1

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]
 

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 ${\href{/padicField/5.12.0.1}{12} }$ R R ${\href{/padicField/13.4.0.1}{4} }^{3}$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.12.0.1}{12} }$ ${\href{/padicField/23.3.0.1}{3} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{3}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.1.0.1}{1} }^{12}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.12.0.1}{12} }$ ${\href{/padicField/59.12.0.1}{12} }$

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
\(2\) Copy content Toggle raw display 2.12.33.375$x^{12} + 80 x^{10} - 264 x^{9} - 638 x^{8} + 64 x^{7} + 208 x^{6} + 3904 x^{5} + 8348 x^{4} + 10496 x^{3} + 13152 x^{2} + 7200 x + 6392$$4$$3$$33$$C_{12}$$[3, 4]^{3}$
\(3\) Copy content Toggle raw display 3.12.6.1$x^{12} + 18 x^{8} + 81 x^{4} - 486 x^{2} + 1458$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
\(7\) Copy content Toggle raw display 7.6.5.5$x^{6} + 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 7$$6$$1$$5$$C_6$$[\ ]_{6}$
\(11\) Copy content Toggle raw display 11.12.6.2$x^{12} + 363 x^{8} - 5324 x^{6} + 87846 x^{4} - 1127357 x^{2} + 3543122$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$