↑

Properties

Label 12.4.46118408000000000.2
Degree $12$
Signature $[4, 4]$
Discriminant $4.612\times 10^{16}$
Root discriminant \(24.47\)
Ramified primes $2,5,7$
Class number $3$
Class group [3]
Galois group $S_3 \times C_4$ (as 12T11)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 10*x^10 + 40*x^9 - 9*x^8 - 150*x^7 + 231*x^6 + 50*x^5 - 409*x^4 + 1190*x^2 - 988*x - 239)
 
gp: K = bnfinit(y^12 - 2*y^11 - 10*y^10 + 40*y^9 - 9*y^8 - 150*y^7 + 231*y^6 + 50*y^5 - 409*y^4 + 1190*y^2 - 988*y - 239, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 - 10*x^10 + 40*x^9 - 9*x^8 - 150*x^7 + 231*x^6 + 50*x^5 - 409*x^4 + 1190*x^2 - 988*x - 239);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^11 - 10*x^10 + 40*x^9 - 9*x^8 - 150*x^7 + 231*x^6 + 50*x^5 - 409*x^4 + 1190*x^2 - 988*x - 239)
 

\( x^{12} - 2 x^{11} - 10 x^{10} + 40 x^{9} - 9 x^{8} - 150 x^{7} + 231 x^{6} + 50 x^{5} - 409 x^{4} + \cdots - 239 \) 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:  $[4, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(46118408000000000\) \(\medspace = 2^{12}\cdot 5^{9}\cdot 7^{8}\) 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:  \(24.47\)
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:  $2\cdot 5^{3/4}7^{2/3}\approx 24.471252165227245$
Ramified primes:   \(2\), \(5\), \(7\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19}a^{10}-\frac{9}{19}a^{9}+\frac{9}{19}a^{8}-\frac{7}{19}a^{7}+\frac{5}{19}a^{6}+\frac{9}{19}a^{5}+\frac{5}{19}a^{4}-\frac{1}{19}a^{3}+\frac{5}{19}a^{2}+\frac{9}{19}a-\frac{5}{19}$, $\frac{1}{217940670248209}a^{11}-\frac{4625925137067}{217940670248209}a^{10}-\frac{84937338751983}{217940670248209}a^{9}-\frac{100123346039555}{217940670248209}a^{8}+\frac{38260375615365}{217940670248209}a^{7}+\frac{30250792944730}{217940670248209}a^{6}-\frac{95914657190791}{217940670248209}a^{5}-\frac{25192079432395}{217940670248209}a^{4}+\frac{44567930063577}{217940670248209}a^{3}+\frac{85963493541424}{217940670248209}a^{2}-\frac{45798522657868}{217940670248209}a-\frac{84965196659337}{217940670248209}$ 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

$C_{3}$, which has order $3$

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:  $7$
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:   $\frac{82092884}{36882834701}a^{11}-\frac{43186535}{36882834701}a^{10}-\frac{815392068}{36882834701}a^{9}+\frac{2102741305}{36882834701}a^{8}+\frac{1482118408}{36882834701}a^{7}-\frac{9215693672}{36882834701}a^{6}+\frac{9139021204}{36882834701}a^{5}+\frac{11170556360}{36882834701}a^{4}-\frac{23638193120}{36882834701}a^{3}-\frac{21277603259}{36882834701}a^{2}+\frac{81290804062}{36882834701}a-\frac{43104138677}{36882834701}$, $\frac{17475858321}{217940670248209}a^{11}+\frac{1344618552}{11470561592011}a^{10}+\frac{421288002831}{217940670248209}a^{9}+\frac{901314853363}{217940670248209}a^{8}-\frac{3746464277275}{217940670248209}a^{7}+\frac{6901908533954}{217940670248209}a^{6}+\frac{22142752123119}{217940670248209}a^{5}-\frac{38506258851407}{217940670248209}a^{4}+\frac{23288872406833}{217940670248209}a^{3}+\frac{156400788376403}{217940670248209}a^{2}-\frac{320990117582}{11470561592011}a-\frac{361250550737181}{217940670248209}$, $\frac{403617656285}{217940670248209}a^{11}-\frac{1433496137250}{217940670248209}a^{10}-\frac{3901678003263}{217940670248209}a^{9}+\frac{23395920735305}{217940670248209}a^{8}-\frac{14503049149479}{217940670248209}a^{7}-\frac{79714257859156}{217940670248209}a^{6}+\frac{152447732262943}{217940670248209}a^{5}-\frac{170267317862}{217940670248209}a^{4}-\frac{228461408739269}{217940670248209}a^{3}+\frac{116937772274086}{217940670248209}a^{2}+\frac{705107517695434}{217940670248209}a-\frac{582795892625368}{217940670248209}$, $\frac{15558126227}{217940670248209}a^{11}+\frac{331611182084}{217940670248209}a^{10}-\frac{21042301896}{11470561592011}a^{9}-\frac{3820273234981}{217940670248209}a^{8}+\frac{7751003071585}{217940670248209}a^{7}+\frac{11689040733972}{217940670248209}a^{6}-\frac{30538566757027}{217940670248209}a^{5}+\frac{1124318639696}{217940670248209}a^{4}+\frac{64782470971995}{217940670248209}a^{3}-\frac{51234860495987}{217940670248209}a^{2}-\frac{225366114417871}{217940670248209}a+\frac{349139568519554}{217940670248209}$, $\frac{134028441972}{217940670248209}a^{11}+\frac{566843259046}{217940670248209}a^{10}-\frac{49929080162}{11470561592011}a^{9}-\frac{2195142238253}{217940670248209}a^{8}+\frac{10360028110910}{217940670248209}a^{7}+\frac{4041671774801}{217940670248209}a^{6}-\frac{20690868713215}{217940670248209}a^{5}+\frac{11426584261167}{217940670248209}a^{4}+\frac{44587560438802}{217940670248209}a^{3}-\frac{73451033763799}{217940670248209}a^{2}-\frac{140431775449066}{217940670248209}a+\frac{37504387499541}{217940670248209}$, $\frac{43782300214}{217940670248209}a^{11}-\frac{97220058622}{217940670248209}a^{10}-\frac{329245463637}{217940670248209}a^{9}+\frac{2022037411367}{217940670248209}a^{8}-\frac{2636399801373}{217940670248209}a^{7}-\frac{5257320870888}{217940670248209}a^{6}+\frac{30242809831758}{217940670248209}a^{5}-\frac{33971527998806}{217940670248209}a^{4}-\frac{100409902787753}{217940670248209}a^{3}+\frac{103556737707063}{217940670248209}a^{2}+\frac{159160093977309}{217940670248209}a-\frac{96458794989793}{217940670248209}$, $\frac{76947956500}{217940670248209}a^{11}-\frac{428168024403}{217940670248209}a^{10}-\frac{1603804001852}{217940670248209}a^{9}+\frac{5174814595467}{217940670248209}a^{8}-\frac{293326886679}{217940670248209}a^{7}-\frac{28589149381331}{217940670248209}a^{6}+\frac{33173126083785}{217940670248209}a^{5}-\frac{10585192244231}{217940670248209}a^{4}-\frac{126400941783811}{217940670248209}a^{3}+\frac{126677066874197}{217940670248209}a^{2}+\frac{30319985071564}{217940670248209}a+\frac{30770714576062}{217940670248209}$ 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:  \( 2782.2583445483015 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 2782.2583445483015 \cdot 3}{2\cdot\sqrt{46118408000000000}}\cr\approx \mathstrut & 0.484608486442020 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 10*x^10 + 40*x^9 - 9*x^8 - 150*x^7 + 231*x^6 + 50*x^5 - 409*x^4 + 1190*x^2 - 988*x - 239)
 
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 - 2*x^11 - 10*x^10 + 40*x^9 - 9*x^8 - 150*x^7 + 231*x^6 + 50*x^5 - 409*x^4 + 1190*x^2 - 988*x - 239, 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 - 2*x^11 - 10*x^10 + 40*x^9 - 9*x^8 - 150*x^7 + 231*x^6 + 50*x^5 - 409*x^4 + 1190*x^2 - 988*x - 239);
 
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 - 2*x^11 - 10*x^10 + 40*x^9 - 9*x^8 - 150*x^7 + 231*x^6 + 50*x^5 - 409*x^4 + 1190*x^2 - 988*x - 239);
 
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

$C_4\times S_3$ (as 12T11):

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 24
The 12 conjugacy class representatives for $S_3 \times C_4$
Character table for $S_3 \times C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.1.980.1, \(\Q(\zeta_{20})^+\), 6.2.4802000.1

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

Galois closure: deg 24
Degree 12 sibling: 12.0.2882400500000000.1
Minimal sibling: 12.0.2882400500000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }$ R R ${\href{/padicField/11.2.0.1}{2} }^{6}$ ${\href{/padicField/13.4.0.1}{4} }^{3}$ ${\href{/padicField/17.4.0.1}{4} }^{3}$ ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.3.0.1}{3} }^{4}$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.4.0.1}{4} }^{3}$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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 2.4.4.2$x^{4} + 4 x^{3} + 4 x^{2} + 12$$2$$2$$4$$C_4$$[2]^{2}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
\(5\) Copy content Toggle raw display 5.4.3.2$x^{4} + 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} + 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} + 5$$4$$1$$3$$C_4$$[\ ]_{4}$
\(7\) Copy content Toggle raw display 7.12.8.2$x^{12} - 70 x^{9} + 1519 x^{6} - 4802 x^{3} + 21609$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.20.2t1.a.a$1$ $ 2^{2} \cdot 5 $ \(\Q(\sqrt{-5}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.20.4t1.a.a$1$ $ 2^{2} \cdot 5 $ \(\Q(\zeta_{20})^+\) $C_4$ (as 4T1) $0$ $1$
1.5.4t1.a.a$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
* 1.20.4t1.a.b$1$ $ 2^{2} \cdot 5 $ \(\Q(\zeta_{20})^+\) $C_4$ (as 4T1) $0$ $1$
1.5.4t1.a.b$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
* 2.980.3t2.a.a$2$ $ 2^{2} \cdot 5 \cdot 7^{2}$ 3.1.980.1 $S_3$ (as 3T2) $1$ $0$
* 2.980.6t3.e.a$2$ $ 2^{2} \cdot 5 \cdot 7^{2}$ 6.0.3841600.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.4900.12t11.a.a$2$ $ 2^{2} \cdot 5^{2} \cdot 7^{2}$ 12.4.46118408000000000.2 $S_3 \times C_4$ (as 12T11) $0$ $0$
* 2.4900.12t11.a.b$2$ $ 2^{2} \cdot 5^{2} \cdot 7^{2}$ 12.4.46118408000000000.2 $S_3 \times C_4$ (as 12T11) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.