Properties

Label 12.12.720...000.1
Degree $12$
Signature $[12, 0]$
Discriminant $7.208\times 10^{20}$
Root discriminant \(54.72\)
Ramified primes $2,3,5,61$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_3^3:(C_4\times S_3)$ (as 12T170)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 39*x^10 - 26*x^9 + 531*x^8 + 708*x^7 - 2653*x^6 - 5778*x^5 + 1089*x^4 + 12320*x^3 + 13176*x^2 + 5856*x + 976)
 
gp: K = bnfinit(y^12 - 39*y^10 - 26*y^9 + 531*y^8 + 708*y^7 - 2653*y^6 - 5778*y^5 + 1089*y^4 + 12320*y^3 + 13176*y^2 + 5856*y + 976, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 39*x^10 - 26*x^9 + 531*x^8 + 708*x^7 - 2653*x^6 - 5778*x^5 + 1089*x^4 + 12320*x^3 + 13176*x^2 + 5856*x + 976);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 39*x^10 - 26*x^9 + 531*x^8 + 708*x^7 - 2653*x^6 - 5778*x^5 + 1089*x^4 + 12320*x^3 + 13176*x^2 + 5856*x + 976)
 

\( x^{12} - 39 x^{10} - 26 x^{9} + 531 x^{8} + 708 x^{7} - 2653 x^{6} - 5778 x^{5} + 1089 x^{4} + 12320 x^{3} + 13176 x^{2} + 5856 x + 976 \) 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:  $[12, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(720795819784500000000\) \(\medspace = 2^{8}\cdot 3^{18}\cdot 5^{9}\cdot 61^{2}\) 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:  \(54.72\)
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:  $2\cdot 3^{313/162}5^{3/4}61^{2/3}\approx 865.6110187539238$
Ramified primes:   \(2\), \(3\), \(5\), \(61\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{8}a^{9}-\frac{1}{4}a^{8}+\frac{1}{8}a^{7}-\frac{1}{2}a^{6}-\frac{1}{8}a^{5}-\frac{1}{4}a^{4}+\frac{3}{8}a^{3}-\frac{3}{8}a-\frac{1}{4}$, $\frac{1}{64}a^{10}-\frac{1}{16}a^{9}-\frac{11}{64}a^{8}-\frac{15}{32}a^{7}+\frac{7}{64}a^{6}-\frac{9}{64}a^{4}+\frac{9}{32}a^{3}+\frac{13}{64}a^{2}+\frac{1}{16}a+\frac{1}{16}$, $\frac{1}{512}a^{11}+\frac{1}{256}a^{10}+\frac{29}{512}a^{9}-\frac{7}{16}a^{8}-\frac{109}{512}a^{7}+\frac{53}{256}a^{6}+\frac{55}{512}a^{5}-\frac{9}{128}a^{4}-\frac{71}{512}a^{3}+\frac{9}{256}a^{2}+\frac{55}{128}a-\frac{13}{64}$ 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 $1$ (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:  $11$
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:   $\frac{89073}{512}a^{11}-\frac{30843}{256}a^{10}-\frac{3431123}{512}a^{9}+\frac{7533}{64}a^{8}+\frac{47255859}{512}a^{7}+\frac{15168753}{256}a^{6}-\frac{257318073}{512}a^{5}-\frac{84115539}{128}a^{4}+\frac{329998713}{512}a^{3}+\frac{434417661}{256}a^{2}+\frac{142987599}{128}a+\frac{15693111}{64}$, $\frac{6561}{512}a^{11}-\frac{2187}{256}a^{10}-\frac{252963}{512}a^{9}-\frac{243}{64}a^{8}+\frac{3485187}{512}a^{7}+\frac{1160865}{256}a^{6}-\frac{18954153}{512}a^{5}-\frac{6318339}{128}a^{4}+\frac{23993833}{512}a^{3}+\frac{32417901}{256}a^{2}+\frac{10805967}{128}a+\frac{1200407}{64}$, $\frac{40095}{512}a^{11}-\frac{13689}{256}a^{10}-\frac{1545021}{512}a^{9}+\frac{783}{32}a^{8}+\frac{21282285}{512}a^{7}+\frac{6927427}{256}a^{6}-\frac{115837911}{512}a^{5}-\frac{38142891}{128}a^{4}+\frac{147873927}{512}a^{3}+\frac{196508511}{256}a^{2}+\frac{64969209}{128}a+\frac{7159893}{64}$, $\frac{5265}{256}a^{11}-\frac{1809}{128}a^{10}-\frac{202851}{256}a^{9}+\frac{627}{64}a^{8}+\frac{2794075}{256}a^{7}+\frac{903767}{128}a^{6}-\frac{15211257}{256}a^{5}-\frac{312005}{4}a^{4}+\frac{19461217}{256}a^{3}+\frac{25747551}{128}a^{2}+\frac{8494283}{64}a+\frac{934201}{32}$, $\frac{17689}{256}a^{11}-\frac{6151}{128}a^{10}-\frac{681291}{256}a^{9}+\frac{433}{8}a^{8}+\frac{9382331}{256}a^{7}+\frac{2999821}{128}a^{6}-\frac{51089537}{256}a^{5}-\frac{16669825}{64}a^{4}+\frac{65570097}{256}a^{3}+\frac{86149185}{128}a^{2}+\frac{28339511}{64}a+\frac{3109603}{32}$, $\frac{21879}{512}a^{11}-\frac{6961}{256}a^{10}-\frac{844485}{512}a^{9}-\frac{1963}{32}a^{8}+\frac{11640117}{512}a^{7}+\frac{4040971}{256}a^{6}-\frac{63220463}{512}a^{5}-\frac{21547387}{128}a^{4}+\frac{78855903}{512}a^{3}+\frac{109748647}{256}a^{2}+\frac{37072417}{128}a+\frac{4167277}{64}$, $\frac{35265}{512}a^{11}-\frac{12375}{256}a^{10}-\frac{1357987}{512}a^{9}+\frac{2267}{32}a^{8}+\frac{18700851}{512}a^{7}+\frac{5920461}{256}a^{6}-\frac{101874505}{512}a^{5}-\frac{33062845}{128}a^{4}+\frac{131253081}{512}a^{3}+\frac{171169425}{256}a^{2}+\frac{56081911}{128}a+\frac{6128299}{64}$, $\frac{3857}{128}a^{11}-\frac{607}{32}a^{10}-\frac{148939}{128}a^{9}-\frac{3227}{64}a^{8}+\frac{2053783}{128}a^{7}+\frac{179551}{16}a^{6}-\frac{11159209}{128}a^{5}-\frac{7628347}{64}a^{4}+\frac{13929629}{128}a^{3}+\frac{9702443}{32}a^{2}+\frac{6542957}{32}a+45807$, $\frac{35109}{256}a^{11}-\frac{12073}{128}a^{10}-\frac{1352639}{256}a^{9}+\frac{4357}{64}a^{8}+\frac{18630695}{256}a^{7}+\frac{6022087}{128}a^{6}-\frac{101425069}{256}a^{5}-\frac{16638239}{32}a^{4}+\frac{129764517}{256}a^{3}+\frac{171647319}{128}a^{2}+\frac{56629023}{64}a+\frac{6228841}{32}$, $\frac{51853}{256}a^{11}-\frac{17993}{128}a^{10}-\frac{1997271}{256}a^{9}+\frac{9473}{64}a^{8}+\frac{27506847}{256}a^{7}+\frac{8811479}{128}a^{6}-\frac{149785653}{256}a^{5}-\frac{24457273}{32}a^{4}+\frac{192195405}{256}a^{3}+\frac{252709783}{128}a^{2}+\frac{83139199}{64}a+\frac{9121281}{32}$, $\frac{230153}{256}a^{11}-\frac{79227}{128}a^{10}-\frac{8866907}{256}a^{9}+\frac{15087}{32}a^{8}+\frac{122129307}{256}a^{7}+\frac{39432257}{128}a^{6}-\frac{664907665}{256}a^{5}-\frac{218012351}{64}a^{4}+\frac{851106929}{256}a^{3}+\frac{1124780189}{128}a^{2}+\frac{370895239}{64}a+\frac{40774919}{32}$ 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:  \( 3902244.46053 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 3902244.46053 \cdot 1}{2\cdot\sqrt{720795819784500000000}}\cr\approx \mathstrut & 0.297672212491 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 39*x^10 - 26*x^9 + 531*x^8 + 708*x^7 - 2653*x^6 - 5778*x^5 + 1089*x^4 + 12320*x^3 + 13176*x^2 + 5856*x + 976)
 
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 - 39*x^10 - 26*x^9 + 531*x^8 + 708*x^7 - 2653*x^6 - 5778*x^5 + 1089*x^4 + 12320*x^3 + 13176*x^2 + 5856*x + 976, 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 - 39*x^10 - 26*x^9 + 531*x^8 + 708*x^7 - 2653*x^6 - 5778*x^5 + 1089*x^4 + 12320*x^3 + 13176*x^2 + 5856*x + 976);
 
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 - 39*x^10 - 26*x^9 + 531*x^8 + 708*x^7 - 2653*x^6 - 5778*x^5 + 1089*x^4 + 12320*x^3 + 13176*x^2 + 5856*x + 976);
 
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

$C_3^3:(C_4\times S_3)$ (as 12T170):

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 648
The 30 conjugacy class representatives for $C_3^3:(C_4\times S_3)$
Character table for $C_3^3:(C_4\times S_3)$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\)

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

Degree 12 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

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 R ${\href{/padicField/7.12.0.1}{12} }$ ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.12.0.1}{12} }$ ${\href{/padicField/17.4.0.1}{4} }^{3}$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.12.0.1}{12} }$ ${\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$

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 2.4.0.1$x^{4} + x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
\(3\) Copy content Toggle raw display 3.12.18.8$x^{12} + 12 x^{11} + 42 x^{10} + 36 x^{9} + 9 x^{8} - 42 x^{6} - 252 x^{5} - 126 x^{4} + 882$$6$$2$$18$12T170$[3/2, 3/2, 2, 2]_{2}^{4}$
\(5\) Copy content Toggle raw display 5.4.3.1$x^{4} + 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
\(61\) Copy content Toggle raw display 61.3.2.1$x^{3} + 61$$3$$1$$2$$C_3$$[\ ]_{3}$
61.3.0.1$x^{3} + 7 x + 59$$1$$3$$0$$C_3$$[\ ]^{3}$
61.3.0.1$x^{3} + 7 x + 59$$1$$3$$0$$C_3$$[\ ]^{3}$
61.3.0.1$x^{3} + 7 x + 59$$1$$3$$0$$C_3$$[\ ]^{3}$