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Properties

Label 12.0.6964478817623209.1
Degree $12$
Signature $[0, 6]$
Discriminant $6.964\times 10^{15}$
Root discriminant \(20.90\)
Ramified primes $19,23$
Class number $4$
Class group [4]
Galois group $C_2 \times S_4$ (as 12T24)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 5*x^11 + 11*x^10 - 9*x^9 - 14*x^8 + 80*x^7 - 225*x^6 + 362*x^5 + 88*x^4 - 1519*x^3 + 2630*x^2 - 1917*x + 581)
 
gp: K = bnfinit(y^12 - 5*y^11 + 11*y^10 - 9*y^9 - 14*y^8 + 80*y^7 - 225*y^6 + 362*y^5 + 88*y^4 - 1519*y^3 + 2630*y^2 - 1917*y + 581, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 5*x^11 + 11*x^10 - 9*x^9 - 14*x^8 + 80*x^7 - 225*x^6 + 362*x^5 + 88*x^4 - 1519*x^3 + 2630*x^2 - 1917*x + 581);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 5*x^11 + 11*x^10 - 9*x^9 - 14*x^8 + 80*x^7 - 225*x^6 + 362*x^5 + 88*x^4 - 1519*x^3 + 2630*x^2 - 1917*x + 581)
 

\( x^{12} - 5 x^{11} + 11 x^{10} - 9 x^{9} - 14 x^{8} + 80 x^{7} - 225 x^{6} + 362 x^{5} + 88 x^{4} + \cdots + 581 \) 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:   \(6964478817623209\) \(\medspace = 19^{6}\cdot 23^{6}\) 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:  \(20.90\)
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:  $19^{1/2}23^{1/2}\approx 20.904544960366874$
Ramified primes:   \(19\), \(23\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{8}a^{9}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{8}a^{3}+\frac{3}{8}a^{2}-\frac{1}{2}a-\frac{3}{8}$, $\frac{1}{16}a^{10}-\frac{1}{16}a^{9}-\frac{1}{8}a^{7}-\frac{3}{8}a^{6}+\frac{3}{8}a^{5}+\frac{1}{16}a^{4}+\frac{1}{4}a^{3}+\frac{1}{16}a^{2}-\frac{7}{16}a+\frac{3}{16}$, $\frac{1}{214991270240}a^{11}-\frac{318680939}{107495635120}a^{10}-\frac{2046692679}{42998254048}a^{9}+\frac{11937637103}{107495635120}a^{8}+\frac{2350484257}{53747817560}a^{7}+\frac{13945436729}{53747817560}a^{6}+\frac{70117187107}{214991270240}a^{5}-\frac{6794972049}{214991270240}a^{4}-\frac{6156578751}{42998254048}a^{3}-\frac{10032925661}{53747817560}a^{2}+\frac{6029241251}{107495635120}a-\frac{37025007703}{214991270240}$ 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

$C_{4}$, which has order $4$

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:   $\frac{353515035}{42998254048}a^{11}-\frac{636102555}{21499127024}a^{10}+\frac{2055946159}{42998254048}a^{9}-\frac{16727407}{21499127024}a^{8}-\frac{1367535889}{10749563512}a^{7}+\frac{5149807193}{10749563512}a^{6}-\frac{49594926799}{42998254048}a^{5}+\frac{54428746593}{42998254048}a^{4}+\frac{119801014403}{42998254048}a^{3}-\frac{98633628715}{10749563512}a^{2}+\frac{181428038171}{21499127024}a-\frac{73309810261}{42998254048}$, $\frac{4086582737}{214991270240}a^{11}-\frac{8459260403}{107495635120}a^{10}+\frac{5527411109}{42998254048}a^{9}-\frac{2153805389}{107495635120}a^{8}-\frac{17944956861}{53747817560}a^{7}+\frac{64686973013}{53747817560}a^{6}-\frac{656183943861}{214991270240}a^{5}+\frac{770140314127}{214991270240}a^{4}+\frac{269771601357}{42998254048}a^{3}-\frac{331094291553}{13436954390}a^{2}+\frac{2561158986367}{107495635120}a-\frac{941263824691}{214991270240}$, $\frac{769238469}{107495635120}a^{11}-\frac{3430201167}{107495635120}a^{10}+\frac{580571687}{10749563512}a^{9}-\frac{363907233}{53747817560}a^{8}-\frac{7239747509}{53747817560}a^{7}+\frac{26736618877}{53747817560}a^{6}-\frac{129961108087}{107495635120}a^{5}+\frac{82521948397}{53747817560}a^{4}+\frac{54386901613}{21499127024}a^{3}-\frac{1137480648681}{107495635120}a^{2}+\frac{1118136288013}{107495635120}a-\frac{54344186441}{53747817560}$, $\frac{1634694237}{214991270240}a^{11}-\frac{827462937}{26873908780}a^{10}+\frac{2349036911}{42998254048}a^{9}-\frac{2828707909}{107495635120}a^{8}-\frac{2828865883}{26873908780}a^{7}+\frac{6630571167}{13436954390}a^{6}-\frac{268191554501}{214991270240}a^{5}+\frac{354081931497}{214991270240}a^{4}+\frac{68506053021}{42998254048}a^{3}-\frac{946399569059}{107495635120}a^{2}+\frac{583234866881}{53747817560}a-\frac{849021447521}{214991270240}$, $\frac{8758050893}{214991270240}a^{11}-\frac{8323136341}{53747817560}a^{10}+\frac{10439386987}{42998254048}a^{9}-\frac{4596760121}{107495635120}a^{8}-\frac{17465984267}{26873908780}a^{7}+\frac{31955143973}{13436954390}a^{6}-\frac{1311855153029}{214991270240}a^{5}+\frac{1432752768533}{214991270240}a^{4}+\frac{546730616917}{42998254048}a^{3}-\frac{4977574580081}{107495635120}a^{2}+\frac{289286124968}{6718477195}a-\frac{3753832402669}{214991270240}$ 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:  \( 552.32724629 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 552.32724629 \cdot 4}{2\cdot\sqrt{6964478817623209}}\cr\approx \mathstrut & 0.81444425164 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 5*x^11 + 11*x^10 - 9*x^9 - 14*x^8 + 80*x^7 - 225*x^6 + 362*x^5 + 88*x^4 - 1519*x^3 + 2630*x^2 - 1917*x + 581)
 
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 - 5*x^11 + 11*x^10 - 9*x^9 - 14*x^8 + 80*x^7 - 225*x^6 + 362*x^5 + 88*x^4 - 1519*x^3 + 2630*x^2 - 1917*x + 581, 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 - 5*x^11 + 11*x^10 - 9*x^9 - 14*x^8 + 80*x^7 - 225*x^6 + 362*x^5 + 88*x^4 - 1519*x^3 + 2630*x^2 - 1917*x + 581);
 
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 - 5*x^11 + 11*x^10 - 9*x^9 - 14*x^8 + 80*x^7 - 225*x^6 + 362*x^5 + 88*x^4 - 1519*x^3 + 2630*x^2 - 1917*x + 581);
 
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

$C_2\times S_4$ (as 12T24):

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 48
The 10 conjugacy class representatives for $C_2 \times S_4$
Character table for $C_2 \times S_4$

Intermediate fields

\(\Q(\sqrt{-19}) \), 3.1.23.1, 6.2.231173.1, 6.0.3628411.1, 6.0.4392287.1

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]
 

Sibling fields

Degree 6 siblings: 6.2.231173.1, 6.0.10051.1
Degree 8 siblings: 8.0.68939809.1, 8.4.36469158961.1
Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed
Minimal sibling: 6.0.10051.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }^{2}$ ${\href{/padicField/3.6.0.1}{6} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{6}$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{6}$ R R ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{6}$

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
\(19\) Copy content Toggle raw display 19.2.1.2$x^{2} + 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.8.4.1$x^{8} + 80 x^{6} + 22 x^{5} + 2250 x^{4} - 792 x^{3} + 25817 x^{2} - 22946 x + 107924$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(23\) Copy content Toggle raw display 23.2.1.2$x^{2} + 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$