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Properties

Label 12.0.164...352.1
Degree $12$
Signature $[0, 6]$
Discriminant $1.647\times 10^{24}$
Root discriminant \(104.24\)
Ramified primes $2,61$
Class number $349$ (GRH)
Class group [349] (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 - 4*x^11 - 62*x^10 + 232*x^9 + 1679*x^8 - 5668*x^7 - 20398*x^6 + 62556*x^5 + 37242*x^4 - 194132*x^3 + 1081520*x^2 - 893952*x + 689057)
 
gp: K = bnfinit(y^12 - 4*y^11 - 62*y^10 + 232*y^9 + 1679*y^8 - 5668*y^7 - 20398*y^6 + 62556*y^5 + 37242*y^4 - 194132*y^3 + 1081520*y^2 - 893952*y + 689057, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - 62*x^10 + 232*x^9 + 1679*x^8 - 5668*x^7 - 20398*x^6 + 62556*x^5 + 37242*x^4 - 194132*x^3 + 1081520*x^2 - 893952*x + 689057);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 - 62*x^10 + 232*x^9 + 1679*x^8 - 5668*x^7 - 20398*x^6 + 62556*x^5 + 37242*x^4 - 194132*x^3 + 1081520*x^2 - 893952*x + 689057)
 

\( x^{12} - 4 x^{11} - 62 x^{10} + 232 x^{9} + 1679 x^{8} - 5668 x^{7} - 20398 x^{6} + 62556 x^{5} + \cdots + 689057 \) 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:   \(1646753279454715263844352\) \(\medspace = 2^{33}\cdot 61^{8}\) 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:  \(104.24\)
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:  $2^{11/4}61^{2/3}\approx 104.24431456608558$
Ramified primes:   \(2\), \(61\) 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:  \(976=2^{4}\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{976}(1,·)$, $\chi_{976}(611,·)$, $\chi_{976}(257,·)$, $\chi_{976}(379,·)$, $\chi_{976}(169,·)$, $\chi_{976}(779,·)$, $\chi_{976}(291,·)$, $\chi_{976}(657,·)$, $\chi_{976}(867,·)$, $\chi_{976}(489,·)$, $\chi_{976}(745,·)$, $\chi_{976}(123,·)$$\rbrace$
This is a CM field.
Reflex fields:  4.0.2048.2$^{2}$, 12.0.1646753279454715263844352.1$^{30}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{7402519347}a^{10}+\frac{270407626}{7402519347}a^{9}+\frac{946349771}{7402519347}a^{8}+\frac{1239941788}{7402519347}a^{7}-\frac{3591468503}{7402519347}a^{6}-\frac{1163401546}{7402519347}a^{5}+\frac{1188125090}{2467506449}a^{4}-\frac{1045951274}{2467506449}a^{3}-\frac{1215930049}{7402519347}a^{2}+\frac{686887660}{2467506449}a-\frac{1156431118}{2467506449}$, $\frac{1}{34\!\cdots\!11}a^{11}-\frac{1033590347}{34\!\cdots\!11}a^{10}+\frac{51\!\cdots\!17}{11\!\cdots\!37}a^{9}-\frac{31\!\cdots\!02}{11\!\cdots\!37}a^{8}-\frac{16\!\cdots\!86}{34\!\cdots\!11}a^{7}+\frac{14\!\cdots\!26}{34\!\cdots\!11}a^{6}+\frac{17\!\cdots\!04}{34\!\cdots\!11}a^{5}+\frac{10\!\cdots\!31}{11\!\cdots\!37}a^{4}-\frac{16\!\cdots\!27}{34\!\cdots\!11}a^{3}+\frac{21\!\cdots\!55}{11\!\cdots\!37}a^{2}-\frac{52\!\cdots\!15}{34\!\cdots\!11}a-\frac{24\!\cdots\!93}{15\!\cdots\!57}$ 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

$C_{349}$, which has order $349$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Relative class number: $349$ (assuming GRH)

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:   $\frac{244}{2467506449}a^{11}+\frac{2610}{2467506449}a^{10}-\frac{28098}{2467506449}a^{9}-\frac{249441}{2467506449}a^{8}+\frac{1336440}{2467506449}a^{7}+\frac{9823418}{2467506449}a^{6}-\frac{31758568}{2467506449}a^{5}-\frac{176535587}{2467506449}a^{4}+\frac{347726016}{2467506449}a^{3}+\frac{1185700715}{2467506449}a^{2}-\frac{1198490624}{2467506449}a+\frac{2204425512}{2467506449}$, $\frac{16508984833484}{34\!\cdots\!11}a^{11}-\frac{598771322495905}{34\!\cdots\!11}a^{10}+\frac{12\!\cdots\!16}{34\!\cdots\!11}a^{9}+\frac{36\!\cdots\!57}{34\!\cdots\!11}a^{8}-\frac{10\!\cdots\!32}{34\!\cdots\!11}a^{7}-\frac{10\!\cdots\!81}{34\!\cdots\!11}a^{6}+\frac{26\!\cdots\!98}{34\!\cdots\!11}a^{5}+\frac{46\!\cdots\!61}{11\!\cdots\!37}a^{4}-\frac{10\!\cdots\!44}{11\!\cdots\!37}a^{3}-\frac{25\!\cdots\!50}{11\!\cdots\!37}a^{2}+\frac{78\!\cdots\!76}{34\!\cdots\!11}a+\frac{95\!\cdots\!58}{15\!\cdots\!57}$, $\frac{269512402444}{50\!\cdots\!19}a^{11}+\frac{29967457173779}{15\!\cdots\!57}a^{10}-\frac{172359495798496}{15\!\cdots\!57}a^{9}-\frac{625492432260913}{50\!\cdots\!19}a^{8}+\frac{80\!\cdots\!56}{15\!\cdots\!57}a^{7}+\frac{17\!\cdots\!81}{50\!\cdots\!19}a^{6}-\frac{54\!\cdots\!70}{50\!\cdots\!19}a^{5}-\frac{68\!\cdots\!87}{15\!\cdots\!57}a^{4}+\frac{14\!\cdots\!36}{15\!\cdots\!57}a^{3}+\frac{23\!\cdots\!94}{15\!\cdots\!57}a^{2}-\frac{89\!\cdots\!52}{50\!\cdots\!19}a+\frac{28\!\cdots\!76}{15\!\cdots\!57}$, $\frac{78721309638092}{34\!\cdots\!11}a^{11}+\frac{320594186562322}{34\!\cdots\!11}a^{10}-\frac{61\!\cdots\!96}{34\!\cdots\!11}a^{9}-\frac{23\!\cdots\!24}{34\!\cdots\!11}a^{8}+\frac{65\!\cdots\!48}{11\!\cdots\!37}a^{7}+\frac{70\!\cdots\!33}{34\!\cdots\!11}a^{6}-\frac{30\!\cdots\!04}{34\!\cdots\!11}a^{5}-\frac{10\!\cdots\!26}{34\!\cdots\!11}a^{4}+\frac{19\!\cdots\!52}{34\!\cdots\!11}a^{3}+\frac{44\!\cdots\!97}{34\!\cdots\!11}a^{2}-\frac{48\!\cdots\!82}{34\!\cdots\!11}a+\frac{67\!\cdots\!81}{50\!\cdots\!19}$, $\frac{33485773538444}{34\!\cdots\!11}a^{11}+\frac{468054866428042}{34\!\cdots\!11}a^{10}-\frac{35\!\cdots\!88}{34\!\cdots\!11}a^{9}-\frac{30\!\cdots\!54}{34\!\cdots\!11}a^{8}+\frac{45\!\cdots\!92}{11\!\cdots\!37}a^{7}+\frac{85\!\cdots\!37}{34\!\cdots\!11}a^{6}-\frac{26\!\cdots\!16}{34\!\cdots\!11}a^{5}-\frac{11\!\cdots\!58}{34\!\cdots\!11}a^{4}+\frac{23\!\cdots\!72}{34\!\cdots\!11}a^{3}+\frac{41\!\cdots\!05}{34\!\cdots\!11}a^{2}-\frac{46\!\cdots\!38}{34\!\cdots\!11}a+\frac{61\!\cdots\!83}{50\!\cdots\!19}$ 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:  \( 27056.6931186 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 27056.6931186 \cdot 349}{2\cdot\sqrt{1646753279454715263844352}}\cr\approx \mathstrut & 0.226378453531 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 62*x^10 + 232*x^9 + 1679*x^8 - 5668*x^7 - 20398*x^6 + 62556*x^5 + 37242*x^4 - 194132*x^3 + 1081520*x^2 - 893952*x + 689057)
 
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 - 62*x^10 + 232*x^9 + 1679*x^8 - 5668*x^7 - 20398*x^6 + 62556*x^5 + 37242*x^4 - 194132*x^3 + 1081520*x^2 - 893952*x + 689057, 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 - 62*x^10 + 232*x^9 + 1679*x^8 - 5668*x^7 - 20398*x^6 + 62556*x^5 + 37242*x^4 - 194132*x^3 + 1081520*x^2 - 893952*x + 689057);
 
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 - 62*x^10 + 232*x^9 + 1679*x^8 - 5668*x^7 - 20398*x^6 + 62556*x^5 + 37242*x^4 - 194132*x^3 + 1081520*x^2 - 893952*x + 689057);
 
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

\(\Q(\sqrt{2}) \), 3.3.3721.1, 4.0.2048.2, 6.6.7089070592.1

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 ${\href{/padicField/3.4.0.1}{4} }^{3}$ ${\href{/padicField/5.12.0.1}{12} }$ ${\href{/padicField/7.3.0.1}{3} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{3}$ ${\href{/padicField/13.12.0.1}{12} }$ ${\href{/padicField/17.3.0.1}{3} }^{4}$ ${\href{/padicField/19.12.0.1}{12} }$ ${\href{/padicField/23.1.0.1}{1} }^{12}$ ${\href{/padicField/29.12.0.1}{12} }$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.12.0.1}{12} }$

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 2.12.33.344$x^{12} + 56 x^{10} - 392 x^{9} + 226 x^{8} - 640 x^{7} - 2480 x^{6} + 3968 x^{5} + 1276 x^{4} - 384 x^{3} + 9280 x^{2} + 24224 x + 31544$$4$$3$$33$$C_{12}$$[3, 4]^{3}$
\(61\) Copy content Toggle raw display 61.12.8.1$x^{12} + 9 x^{10} + 364 x^{9} + 33 x^{8} + 720 x^{7} - 16731 x^{6} - 1002 x^{5} - 64041 x^{4} - 1125646 x^{3} + 428523 x^{2} - 4549998 x + 62837938$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$