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Properties

Label 12.2.449...576.10
Degree $12$
Signature $[2, 5]$
Discriminant $-4.492\times 10^{18}$
Root discriminant \(35.84\)
Ramified primes $2,17$
Class number $2$
Class group [2]
Galois group $S_4^2:D_4$ (as 12T260)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 4*x^10 - 20*x^9 - 9*x^8 + 25*x^7 - 14*x^6 - 322*x^5 + 304*x^4 - 1200*x^3 + 880*x^2 - 920*x + 608)
 
gp: K = bnfinit(y^12 - y^11 + 4*y^10 - 20*y^9 - 9*y^8 + 25*y^7 - 14*y^6 - 322*y^5 + 304*y^4 - 1200*y^3 + 880*y^2 - 920*y + 608, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 + 4*x^10 - 20*x^9 - 9*x^8 + 25*x^7 - 14*x^6 - 322*x^5 + 304*x^4 - 1200*x^3 + 880*x^2 - 920*x + 608);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 + 4*x^10 - 20*x^9 - 9*x^8 + 25*x^7 - 14*x^6 - 322*x^5 + 304*x^4 - 1200*x^3 + 880*x^2 - 920*x + 608)
 

\( x^{12} - x^{11} + 4 x^{10} - 20 x^{9} - 9 x^{8} + 25 x^{7} - 14 x^{6} - 322 x^{5} + 304 x^{4} + \cdots + 608 \) 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:  $[2, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-4492085992834072576\) \(\medspace = -\,2^{17}\cdot 17^{11}\) 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:  \(35.84\)
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:  $2^{51/16}17^{11/12}\approx 122.30511559717738$
Ramified primes:   \(2\), \(17\) 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:  \(\Q(\sqrt{-34}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{12}a^{9}-\frac{1}{12}a^{8}-\frac{1}{6}a^{7}-\frac{5}{12}a^{5}-\frac{1}{4}a^{4}+\frac{1}{3}a^{3}-\frac{1}{6}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{36}a^{10}-\frac{1}{12}a^{8}+\frac{1}{9}a^{7}+\frac{7}{36}a^{6}+\frac{1}{9}a^{5}+\frac{13}{36}a^{4}-\frac{1}{9}a^{3}+\frac{1}{18}a^{2}-\frac{4}{9}$, $\frac{1}{3699654022728}a^{11}-\frac{34107734423}{3699654022728}a^{10}+\frac{9912411905}{308304501894}a^{9}+\frac{7752388595}{1849827011364}a^{8}+\frac{139987159379}{3699654022728}a^{7}-\frac{577596839653}{3699654022728}a^{6}-\frac{120738858995}{1849827011364}a^{5}-\frac{43780691996}{154152250947}a^{4}-\frac{180311344985}{924913505682}a^{3}+\frac{214118093362}{462456752841}a^{2}-\frac{93569840606}{462456752841}a-\frac{64999448600}{462456752841}$ 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:  $6$
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:   $\frac{3165278492}{462456752841}a^{11}+\frac{3286874672}{462456752841}a^{10}+\frac{1829419145}{308304501894}a^{9}-\frac{84503815709}{924913505682}a^{8}-\frac{316445233579}{924913505682}a^{7}+\frac{141531810167}{924913505682}a^{6}+\frac{627020986013}{924913505682}a^{5}-\frac{768720816233}{308304501894}a^{4}-\frac{3442074281327}{924913505682}a^{3}-\frac{1335618667723}{924913505682}a^{2}-\frac{3109737735761}{462456752841}a+\frac{3194871404857}{462456752841}$, $\frac{5065724888}{462456752841}a^{11}-\frac{6821501885}{924913505682}a^{10}+\frac{7034734475}{308304501894}a^{9}-\frac{87987067525}{462456752841}a^{8}-\frac{195746403403}{924913505682}a^{7}+\frac{258229185358}{462456752841}a^{6}+\frac{200038454663}{924913505682}a^{5}-\frac{686872680133}{154152250947}a^{4}+\frac{1811993131153}{924913505682}a^{3}-\frac{4683683239315}{924913505682}a^{2}+\frac{348539876611}{462456752841}a+\frac{1020464258599}{462456752841}$, $\frac{2168212481}{1849827011364}a^{11}+\frac{958040945}{1849827011364}a^{10}+\frac{1250453663}{308304501894}a^{9}-\frac{7754235352}{462456752841}a^{8}-\frac{67929346631}{1849827011364}a^{7}-\frac{65153900531}{1849827011364}a^{6}+\frac{4417235899}{462456752841}a^{5}-\frac{64394498339}{102768167298}a^{4}+\frac{282269120761}{924913505682}a^{3}-\frac{196629527719}{924913505682}a^{2}+\frac{75971130907}{462456752841}a+\frac{323502699469}{462456752841}$, $\frac{38136705889}{924913505682}a^{11}-\frac{73778123665}{1849827011364}a^{10}+\frac{68186121871}{616609003788}a^{9}-\frac{356348417461}{462456752841}a^{8}-\frac{250458678977}{462456752841}a^{7}+\frac{3693588470305}{1849827011364}a^{6}+\frac{329833259695}{1849827011364}a^{5}-\frac{542490014085}{34256055766}a^{4}+\frac{5474428048283}{462456752841}a^{3}-\frac{13426014513290}{462456752841}a^{2}+\frac{9477355703503}{462456752841}a-\frac{226840594775}{462456752841}$, $\frac{5001581107}{1849827011364}a^{11}-\frac{5183493809}{462456752841}a^{10}+\frac{7205771243}{616609003788}a^{9}-\frac{34516067222}{462456752841}a^{8}+\frac{214264949993}{1849827011364}a^{7}+\frac{154182159425}{462456752841}a^{6}-\frac{701213524015}{1849827011364}a^{5}-\frac{154047362356}{154152250947}a^{4}+\frac{3382877811767}{924913505682}a^{3}-\frac{1222527810088}{462456752841}a^{2}+\frac{3441966062585}{462456752841}a-\frac{726227982541}{462456752841}$, $\frac{363921674}{462456752841}a^{11}-\frac{1276137433}{462456752841}a^{10}+\frac{905258057}{308304501894}a^{9}-\frac{21930301241}{924913505682}a^{8}+\frac{6813649588}{462456752841}a^{7}+\frac{19787070712}{462456752841}a^{6}-\frac{75845379325}{924913505682}a^{5}-\frac{63659557393}{308304501894}a^{4}+\frac{129341567534}{462456752841}a^{3}-\frac{227390475620}{462456752841}a^{2}+\frac{351992505214}{462456752841}a-\frac{90841478591}{462456752841}$ 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:  \( 128565.789094 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 128565.789094 \cdot 2}{2\cdot\sqrt{4492085992834072576}}\cr\approx \mathstrut & 2.37607815811 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 4*x^10 - 20*x^9 - 9*x^8 + 25*x^7 - 14*x^6 - 322*x^5 + 304*x^4 - 1200*x^3 + 880*x^2 - 920*x + 608)
 
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 - x^11 + 4*x^10 - 20*x^9 - 9*x^8 + 25*x^7 - 14*x^6 - 322*x^5 + 304*x^4 - 1200*x^3 + 880*x^2 - 920*x + 608, 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 - x^11 + 4*x^10 - 20*x^9 - 9*x^8 + 25*x^7 - 14*x^6 - 322*x^5 + 304*x^4 - 1200*x^3 + 880*x^2 - 920*x + 608);
 
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 - x^11 + 4*x^10 - 20*x^9 - 9*x^8 + 25*x^7 - 14*x^6 - 322*x^5 + 304*x^4 - 1200*x^3 + 880*x^2 - 920*x + 608);
 
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

$S_4^2:D_4$ (as 12T260):

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 4608
The 65 conjugacy class representatives for $S_4^2:D_4$
Character table for $S_4^2:D_4$

Intermediate fields

\(\Q(\sqrt{17}) \), 6.2.11358856.1

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 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 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 ${\href{/padicField/3.4.0.1}{4} }^{3}$ ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{3}$ ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ R ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.4.11.15$x^{4} + 12 x^{2} + 8 x + 2$$4$$1$$11$$D_{4}$$[2, 3, 4]$
2.6.6.1$x^{6} + 6 x^{5} + 34 x^{4} + 80 x^{3} + 204 x^{2} + 216 x + 216$$2$$3$$6$$A_4$$[2, 2]^{3}$
\(17\) Copy content Toggle raw display 17.12.11.2$x^{12} + 34$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$