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Properties

Label 12.0.333621760000000000.2
Degree $12$
Signature $(0, 6)$
Discriminant $3.336\times 10^{17}$
Root discriminant \(28.86\)
Ramified primes $2,5,19$
Class number $3$
Class group [3]
Galois group $C_6\times S_3$ (as 12T18)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 17*x^10 - 20*x^9 - 70*x^8 + 224*x^7 - 164*x^6 + 238*x^5 - 120*x^4 - 1480*x^3 + 2012*x^2 - 672*x + 2339)
 
Copy content gp:K = bnfinit(y^12 - 6*y^11 + 17*y^10 - 20*y^9 - 70*y^8 + 224*y^7 - 164*y^6 + 238*y^5 - 120*y^4 - 1480*y^3 + 2012*y^2 - 672*y + 2339, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 + 17*x^10 - 20*x^9 - 70*x^8 + 224*x^7 - 164*x^6 + 238*x^5 - 120*x^4 - 1480*x^3 + 2012*x^2 - 672*x + 2339);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 6*x^11 + 17*x^10 - 20*x^9 - 70*x^8 + 224*x^7 - 164*x^6 + 238*x^5 - 120*x^4 - 1480*x^3 + 2012*x^2 - 672*x + 2339)
 

\( x^{12} - 6 x^{11} + 17 x^{10} - 20 x^{9} - 70 x^{8} + 224 x^{7} - 164 x^{6} + 238 x^{5} - 120 x^{4} + \cdots + 2339 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $12$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(0, 6)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(333621760000000000\) \(\medspace = 2^{18}\cdot 5^{10}\cdot 19^{4}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(28.86\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{3/2}5^{5/6}19^{2/3}\approx 77.00561572177682$
Ramified primes:   \(2\), \(5\), \(19\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_6$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\sqrt{-2}, \sqrt{5})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{11}a^{8}-\frac{5}{11}a^{7}-\frac{5}{11}a^{6}-\frac{1}{11}a^{5}+\frac{1}{11}a^{4}+\frac{3}{11}a^{3}-\frac{1}{11}a^{2}+\frac{3}{11}a+\frac{4}{11}$, $\frac{1}{11}a^{9}+\frac{3}{11}a^{7}-\frac{4}{11}a^{6}-\frac{4}{11}a^{5}-\frac{3}{11}a^{4}+\frac{3}{11}a^{3}-\frac{2}{11}a^{2}-\frac{3}{11}a-\frac{2}{11}$, $\frac{1}{33}a^{10}-\frac{1}{33}a^{9}+\frac{1}{33}a^{8}+\frac{14}{33}a^{7}+\frac{10}{33}a^{6}+\frac{1}{11}a^{5}-\frac{7}{33}a^{4}+\frac{1}{3}a^{3}+\frac{4}{11}a^{2}-\frac{5}{33}a+\frac{16}{33}$, $\frac{1}{10782024513447}a^{11}-\frac{9246437506}{10782024513447}a^{10}-\frac{21016635100}{980184046677}a^{9}-\frac{70353909349}{10782024513447}a^{8}-\frac{4312942396619}{10782024513447}a^{7}+\frac{1106855397300}{3594008171149}a^{6}-\frac{4318190359111}{10782024513447}a^{5}-\frac{1830809735335}{10782024513447}a^{4}-\frac{1702905460760}{3594008171149}a^{3}-\frac{6265352843}{10782024513447}a^{2}+\frac{4277742392929}{10782024513447}a-\frac{1163105662783}{3594008171149}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $11$

Class group and class number

Ideal class group:  $C_{3}$, which has order $3$
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{3}$, which has order $3$
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $5$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{21790898}{82305530637}a^{11}-\frac{69470077}{82305530637}a^{10}+\frac{2389088}{7482320967}a^{9}+\frac{638210039}{82305530637}a^{8}-\frac{2904697766}{82305530637}a^{7}+\frac{890411497}{82305530637}a^{6}+\frac{10027625086}{82305530637}a^{5}-\frac{4520271780}{27435176879}a^{4}+\frac{6378840956}{82305530637}a^{3}-\frac{13711906048}{82305530637}a^{2}+\frac{502739738}{27435176879}a+\frac{136934512312}{82305530637}$, $\frac{5752687343}{10782024513447}a^{11}-\frac{53639627500}{10782024513447}a^{10}+\frac{20855429009}{980184046677}a^{9}-\frac{620236248145}{10782024513447}a^{8}+\frac{862290961264}{10782024513447}a^{7}+\frac{119169511039}{10782024513447}a^{6}-\frac{1610378277434}{10782024513447}a^{5}+\frac{1996946406874}{3594008171149}a^{4}-\frac{15934004980852}{10782024513447}a^{3}+\frac{10933458103796}{10782024513447}a^{2}-\frac{3887994281328}{3594008171149}a+\frac{8988351453838}{10782024513447}$, $\frac{3749432164}{10782024513447}a^{11}-\frac{17673308632}{10782024513447}a^{10}+\frac{8477429387}{980184046677}a^{9}-\frac{539658880399}{10782024513447}a^{8}+\frac{1998773383216}{10782024513447}a^{7}-\frac{1916909678367}{3594008171149}a^{6}+\frac{7634685015596}{10782024513447}a^{5}+\frac{7278138744095}{10782024513447}a^{4}-\frac{9531667905475}{3594008171149}a^{3}+\frac{49803386531416}{10782024513447}a^{2}-\frac{110431976798030}{10782024513447}a+\frac{41005247259514}{3594008171149}$, $\frac{2727922154}{10782024513447}a^{11}-\frac{42407848750}{3594008171149}a^{10}+\frac{20155496327}{326728015559}a^{9}-\frac{745710209797}{3594008171149}a^{8}+\frac{1366587400292}{3594008171149}a^{7}-\frac{120175636294}{10782024513447}a^{6}-\frac{6495547368422}{10782024513447}a^{5}+\frac{17773441305941}{10782024513447}a^{4}-\frac{51310797116969}{10782024513447}a^{3}+\frac{67537571938439}{10782024513447}a^{2}-\frac{31900628933498}{10782024513447}a+\frac{62882338447187}{10782024513447}$, $\frac{103657977367}{10782024513447}a^{11}-\frac{831867350657}{10782024513447}a^{10}+\frac{309597530833}{980184046677}a^{9}-\frac{8725799854235}{10782024513447}a^{8}+\frac{9617756047316}{10782024513447}a^{7}+\frac{5390897023181}{10782024513447}a^{6}-\frac{28279586770642}{10782024513447}a^{5}+\frac{26843406084025}{3594008171149}a^{4}-\frac{173536855629638}{10782024513447}a^{3}+\frac{172072808400550}{10782024513447}a^{2}-\frac{34293662301446}{3594008171149}a+\frac{127398532284578}{10782024513447}$ Copy content Toggle raw display
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 4623.362856545208 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 4623.362856545208 \cdot 3}{2\cdot\sqrt{333621760000000000}}\cr\approx \mathstrut & 0.738756413561221 \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 17*x^10 - 20*x^9 - 70*x^8 + 224*x^7 - 164*x^6 + 238*x^5 - 120*x^4 - 1480*x^3 + 2012*x^2 - 672*x + 2339) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^12 - 6*x^11 + 17*x^10 - 20*x^9 - 70*x^8 + 224*x^7 - 164*x^6 + 238*x^5 - 120*x^4 - 1480*x^3 + 2012*x^2 - 672*x + 2339, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 + 17*x^10 - 20*x^9 - 70*x^8 + 224*x^7 - 164*x^6 + 238*x^5 - 120*x^4 - 1480*x^3 + 2012*x^2 - 672*x + 2339); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 6*x^11 + 17*x^10 - 20*x^9 - 70*x^8 + 224*x^7 - 164*x^6 + 238*x^5 - 120*x^4 - 1480*x^3 + 2012*x^2 - 672*x + 2339); 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

$C_6\times S_3$ (as 12T18):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 36
The 18 conjugacy class representatives for $C_6\times S_3$
Character table for $C_6\times S_3$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-2}, \sqrt{5})\), 6.0.115520000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Galois closure: deg 36
Degree 18 siblings: 18.0.9065737908494995456000000000000000.1, deg 18
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 ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{3}$ R ${\href{/padicField/7.2.0.1}{2} }^{6}$ ${\href{/padicField/11.1.0.1}{1} }^{12}$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ R ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.2.0.1}{2} }^{6}$ ${\href{/padicField/41.3.0.1}{3} }^{4}$ ${\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 2.6.2.18a1.1$x^{12} + 2 x^{10} + 2 x^{9} + x^{8} + 4 x^{7} + 3 x^{6} + 2 x^{5} + 4 x^{4} + 2 x^{3} + x^{2} + 2 x + 3$$2$$6$$18$$C_6\times C_2$$$[3]^{6}$$
\(5\) Copy content Toggle raw display 5.2.6.10a1.3$x^{12} + 24 x^{11} + 252 x^{10} + 1520 x^{9} + 5820 x^{8} + 14784 x^{7} + 25376 x^{6} + 29568 x^{5} + 23280 x^{4} + 12160 x^{3} + 4032 x^{2} + 773 x + 79$$6$$2$$10$$C_6\times S_3$$$[\ ]_{6}^{6}$$
\(19\) Copy content Toggle raw display 19.3.1.0a1.1$x^{3} + 4 x + 17$$1$$3$$0$$C_3$$$[\ ]^{3}$$
19.1.3.2a1.2$x^{3} + 38$$3$$1$$2$$C_3$$$[\ ]_{3}$$
19.3.1.0a1.1$x^{3} + 4 x + 17$$1$$3$$0$$C_3$$$[\ ]^{3}$$
19.1.3.2a1.2$x^{3} + 38$$3$$1$$2$$C_3$$$[\ ]_{3}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*36 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
*36 1.40.2t1.b.a$1$ $ 2^{3} \cdot 5 $ \(\Q(\sqrt{-10}) \) $C_2$ (as 2T1) $1$ $-1$
*36 1.8.2t1.b.a$1$ $ 2^{3}$ \(\Q(\sqrt{-2}) \) $C_2$ (as 2T1) $1$ $-1$
*36 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.152.6t1.c.a$1$ $ 2^{3} \cdot 19 $ 6.0.66724352.1 $C_6$ (as 6T1) $0$ $-1$
1.95.6t1.a.a$1$ $ 5 \cdot 19 $ 6.6.16290125.1 $C_6$ (as 6T1) $0$ $1$
1.19.3t1.a.a$1$ $ 19 $ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
1.19.3t1.a.b$1$ $ 19 $ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
1.760.6t1.b.a$1$ $ 2^{3} \cdot 5 \cdot 19 $ 6.0.8340544000.3 $C_6$ (as 6T1) $0$ $-1$
1.152.6t1.c.b$1$ $ 2^{3} \cdot 19 $ 6.0.66724352.1 $C_6$ (as 6T1) $0$ $-1$
1.760.6t1.b.b$1$ $ 2^{3} \cdot 5 \cdot 19 $ 6.0.8340544000.3 $C_6$ (as 6T1) $0$ $-1$
1.95.6t1.a.b$1$ $ 5 \cdot 19 $ 6.6.16290125.1 $C_6$ (as 6T1) $0$ $1$
2.72200.3t2.a.a$2$ $ 2^{3} \cdot 5^{2} \cdot 19^{2}$ 3.1.72200.2 $S_3$ (as 3T2) $1$ $0$
2.72200.6t3.e.a$2$ $ 2^{3} \cdot 5^{2} \cdot 19^{2}$ 6.0.208513600000.4 $D_{6}$ (as 6T3) $1$ $0$
*36 2.3800.6t5.c.a$2$ $ 2^{3} \cdot 5^{2} \cdot 19 $ 6.0.115520000.1 $S_3\times C_3$ (as 6T5) $0$ $0$
*36 2.3800.12t18.e.a$2$ $ 2^{3} \cdot 5^{2} \cdot 19 $ 12.0.333621760000000000.2 $C_6\times S_3$ (as 12T18) $0$ $0$
*36 2.3800.6t5.c.b$2$ $ 2^{3} \cdot 5^{2} \cdot 19 $ 6.0.115520000.1 $S_3\times C_3$ (as 6T5) $0$ $0$
*36 2.3800.12t18.e.b$2$ $ 2^{3} \cdot 5^{2} \cdot 19 $ 12.0.333621760000000000.2 $C_6\times S_3$ (as 12T18) $0$ $0$

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 *.

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)