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Properties

Label 12.12.308...577.1
Degree $12$
Signature $[12, 0]$
Discriminant $3.085\times 10^{19}$
Root discriminant \(42.08\)
Ramified primes $17,127$
Class number $1$
Class group trivial
Galois group $C_3\times (C_3 : C_4)$ (as 12T19)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 29*x^10 + 20*x^9 + 283*x^8 - 117*x^7 - 1027*x^6 + 249*x^5 + 1081*x^4 - 241*x^3 - 279*x^2 + 120*x - 13)
 
gp: K = bnfinit(y^12 - y^11 - 29*y^10 + 20*y^9 + 283*y^8 - 117*y^7 - 1027*y^6 + 249*y^5 + 1081*y^4 - 241*y^3 - 279*y^2 + 120*y - 13, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 29*x^10 + 20*x^9 + 283*x^8 - 117*x^7 - 1027*x^6 + 249*x^5 + 1081*x^4 - 241*x^3 - 279*x^2 + 120*x - 13);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 29*x^10 + 20*x^9 + 283*x^8 - 117*x^7 - 1027*x^6 + 249*x^5 + 1081*x^4 - 241*x^3 - 279*x^2 + 120*x - 13)
 

\( x^{12} - x^{11} - 29 x^{10} + 20 x^{9} + 283 x^{8} - 117 x^{7} - 1027 x^{6} + 249 x^{5} + 1081 x^{4} + \cdots - 13 \) 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:   \(30850000558264402577\) \(\medspace = 17^{9}\cdot 127^{4}\) 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:  \(42.08\)
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:  $17^{3/4}127^{2/3}\approx 211.53026087633702$
Ramified primes:   \(17\), \(127\) 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:  \(\Q(\sqrt{17}) \)
$\card{ \Aut(K/\Q) }$:  $6$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{31263178}a^{11}+\frac{3275991}{31263178}a^{10}+\frac{2774669}{31263178}a^{9}+\frac{8811579}{31263178}a^{8}-\frac{5515581}{31263178}a^{7}+\frac{3904856}{15631589}a^{6}-\frac{11577981}{31263178}a^{5}+\frac{2709135}{15631589}a^{4}-\frac{3620097}{15631589}a^{3}-\frac{7211293}{31263178}a^{2}+\frac{7330477}{31263178}a-\frac{3665075}{15631589}$ 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 $1$

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:   $\frac{29903716}{15631589}a^{11}-\frac{20518030}{15631589}a^{10}-\frac{873695472}{15631589}a^{9}+\frac{323946008}{15631589}a^{8}+\frac{8565750318}{15631589}a^{7}-\frac{813025594}{15631589}a^{6}-\frac{30978725412}{15631589}a^{5}-\frac{2250119387}{15631589}a^{4}+\frac{31664306134}{15631589}a^{3}+\frac{2644448845}{15631589}a^{2}-\frac{7557466814}{15631589}a+\frac{1266708454}{15631589}$, $\frac{52087134}{15631589}a^{11}-\frac{38167980}{15631589}a^{10}-\frac{1520619884}{15631589}a^{9}+\frac{635331147}{15631589}a^{8}+\frac{14907341181}{15631589}a^{7}-\frac{2109211218}{15631589}a^{6}-\frac{54027528825}{15631589}a^{5}-\frac{1478748531}{15631589}a^{4}+\frac{55813076286}{15631589}a^{3}+\frac{2401280940}{15631589}a^{2}-\frac{13835512481}{15631589}a+\frac{2527934442}{15631589}$, $\frac{29903716}{15631589}a^{11}-\frac{20518030}{15631589}a^{10}-\frac{873695472}{15631589}a^{9}+\frac{323946008}{15631589}a^{8}+\frac{8565750318}{15631589}a^{7}-\frac{813025594}{15631589}a^{6}-\frac{30978725412}{15631589}a^{5}-\frac{2250119387}{15631589}a^{4}+\frac{31664306134}{15631589}a^{3}+\frac{2644448845}{15631589}a^{2}-\frac{7541835225}{15631589}a+\frac{1266708454}{15631589}$, $\frac{7170953}{31263178}a^{11}-\frac{1466997}{15631589}a^{10}-\frac{209806297}{31263178}a^{9}+\frac{9298783}{15631589}a^{8}+\frac{2044371017}{31263178}a^{7}+\frac{393417207}{31263178}a^{6}-\frac{3589809157}{15631589}a^{5}-\frac{1344054500}{15631589}a^{4}+\frac{3195201494}{15631589}a^{3}+\frac{2830506387}{31263178}a^{2}-\frac{365069842}{15631589}a-\frac{80348804}{15631589}$, $\frac{119832423}{31263178}a^{11}-\frac{41523303}{15631589}a^{10}-\frac{3501272919}{31263178}a^{9}+\frac{661459750}{15631589}a^{8}+\frac{34335802023}{31263178}a^{7}-\frac{3504641147}{31263178}a^{6}-\frac{62147072557}{15631589}a^{5}-\frac{4070547651}{15631589}a^{4}+\frac{63747962124}{15631589}a^{3}+\frac{9825846895}{31263178}a^{2}-\frac{15318955326}{15631589}a+\frac{2593246229}{15631589}$, $\frac{119832423}{31263178}a^{11}-\frac{41523303}{15631589}a^{10}-\frac{3501272919}{31263178}a^{9}+\frac{661459750}{15631589}a^{8}+\frac{34335802023}{31263178}a^{7}-\frac{3504641147}{31263178}a^{6}-\frac{62147072557}{15631589}a^{5}-\frac{4070547651}{15631589}a^{4}+\frac{63747962124}{15631589}a^{3}+\frac{9825846895}{31263178}a^{2}-\frac{15334586915}{15631589}a+\frac{2593246229}{15631589}$, $\frac{51537789}{31263178}a^{11}-\frac{19116947}{15631589}a^{10}-\frac{1503655121}{31263178}a^{9}+\frac{320683922}{15631589}a^{8}+\frac{14727552743}{31263178}a^{7}-\frac{2198954041}{31263178}a^{6}-\frac{26638612570}{15631589}a^{5}-\frac{572683644}{15631589}a^{4}+\frac{27343971646}{15631589}a^{3}+\frac{2344170577}{31263178}a^{2}-\frac{6643115509}{15631589}a+\frac{1196508773}{15631589}$, $\frac{41234451}{15631589}a^{11}-\frac{31137605}{15631589}a^{10}-\frac{2407998341}{31263178}a^{9}+\frac{530529333}{15631589}a^{8}+\frac{23631311697}{31263178}a^{7}-\frac{1945835533}{15631589}a^{6}-\frac{85956826291}{31263178}a^{5}-\frac{307563543}{31263178}a^{4}+\frac{45045983421}{15631589}a^{3}+\frac{835635898}{15631589}a^{2}-\frac{11659311157}{15631589}a+\frac{4469765909}{31263178}$, $\frac{2027547}{332587}a^{11}-\frac{2912511}{665174}a^{10}-\frac{118382385}{665174}a^{9}+\frac{47680763}{665174}a^{8}+\frac{1160014045}{665174}a^{7}-\frac{146130441}{665174}a^{6}-\frac{2098183371}{332587}a^{5}-\frac{182962609}{665174}a^{4}+\frac{2150977960}{332587}a^{3}+\frac{130144252}{332587}a^{2}-\frac{1040671817}{665174}a+\frac{180260617}{665174}$, $\frac{230451815}{31263178}a^{11}-\frac{156795775}{31263178}a^{10}-\frac{3364624707}{15631589}a^{9}+\frac{2452192007}{31263178}a^{8}+\frac{32945750624}{15631589}a^{7}-\frac{2883778238}{15631589}a^{6}-\frac{118757005391}{15631589}a^{5}-\frac{19478250669}{31263178}a^{4}+\frac{119875091575}{15631589}a^{3}+\frac{23451981941}{31263178}a^{2}-\frac{55061178305}{31263178}a+\frac{8758264623}{31263178}$, $\frac{1427097}{31263178}a^{11}-\frac{1236349}{31263178}a^{10}-\frac{41056409}{31263178}a^{9}+\frac{21132401}{31263178}a^{8}+\frac{392700729}{31263178}a^{7}-\frac{37563879}{15631589}a^{6}-\frac{1344063031}{31263178}a^{5}-\frac{25002631}{15631589}a^{4}+\frac{557701706}{15631589}a^{3}+\frac{185906687}{31263178}a^{2}-\frac{76676625}{31263178}a-\frac{9699930}{15631589}$ 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:  \( 756757.5012453796 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 756757.5012453796 \cdot 1}{2\cdot\sqrt{30850000558264402577}}\cr\approx \mathstrut & 0.279035268228827 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 29*x^10 + 20*x^9 + 283*x^8 - 117*x^7 - 1027*x^6 + 249*x^5 + 1081*x^4 - 241*x^3 - 279*x^2 + 120*x - 13)
 
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 - 29*x^10 + 20*x^9 + 283*x^8 - 117*x^7 - 1027*x^6 + 249*x^5 + 1081*x^4 - 241*x^3 - 279*x^2 + 120*x - 13, 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 - 29*x^10 + 20*x^9 + 283*x^8 - 117*x^7 - 1027*x^6 + 249*x^5 + 1081*x^4 - 241*x^3 - 279*x^2 + 120*x - 13);
 
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 - 29*x^10 + 20*x^9 + 283*x^8 - 117*x^7 - 1027*x^6 + 249*x^5 + 1081*x^4 - 241*x^3 - 279*x^2 + 120*x - 13);
 
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

$C_3:C_{12}$ (as 12T19):

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 36
The 18 conjugacy class representatives for $C_3\times (C_3 : C_4)$
Character table for $C_3\times (C_3 : C_4)$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 6.6.79241777.1

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 36
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 ${\href{/padicField/2.6.0.1}{6} }^{2}$ ${\href{/padicField/3.12.0.1}{12} }$ ${\href{/padicField/5.4.0.1}{4} }^{3}$ ${\href{/padicField/7.12.0.1}{12} }$ ${\href{/padicField/11.12.0.1}{12} }$ ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{6}$ R ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.12.0.1}{12} }$ ${\href{/padicField/31.12.0.1}{12} }$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ ${\href{/padicField/47.1.0.1}{1} }^{12}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$

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
\(17\) Copy content Toggle raw display 17.12.9.1$x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
\(127\) Copy content Toggle raw display 127.6.4.2$x^{6} - 16002 x^{3} + 48387$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
127.6.0.1$x^{6} + 84 x^{3} + 115 x^{2} + 82 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.17.2t1.a.a$1$ $ 17 $ \(\Q(\sqrt{17}) \) $C_2$ (as 2T1) $1$ $1$
1.2159.6t1.a.a$1$ $ 17 \cdot 127 $ 6.6.1278090621233.3 $C_6$ (as 6T1) $0$ $1$
1.127.3t1.a.a$1$ $ 127 $ 3.3.16129.1 $C_3$ (as 3T1) $0$ $1$
1.127.3t1.a.b$1$ $ 127 $ 3.3.16129.1 $C_3$ (as 3T1) $0$ $1$
1.2159.6t1.a.b$1$ $ 17 \cdot 127 $ 6.6.1278090621233.3 $C_6$ (as 6T1) $0$ $1$
* 1.17.4t1.a.a$1$ $ 17 $ 4.4.4913.1 $C_4$ (as 4T1) $0$ $1$
* 1.17.4t1.a.b$1$ $ 17 $ 4.4.4913.1 $C_4$ (as 4T1) $0$ $1$
1.2159.12t1.a.a$1$ $ 17 \cdot 127 $ 12.12.8025462320079492591473139857.2 $C_{12}$ (as 12T1) $0$ $1$
1.2159.12t1.a.b$1$ $ 17 \cdot 127 $ 12.12.8025462320079492591473139857.2 $C_{12}$ (as 12T1) $0$ $1$
1.2159.12t1.a.c$1$ $ 17 \cdot 127 $ 12.12.8025462320079492591473139857.2 $C_{12}$ (as 12T1) $0$ $1$
1.2159.12t1.a.d$1$ $ 17 \cdot 127 $ 12.12.8025462320079492591473139857.2 $C_{12}$ (as 12T1) $0$ $1$
2.274193.3t2.a.a$2$ $ 17 \cdot 127^{2}$ 3.3.274193.1 $S_3$ (as 3T2) $1$ $2$
2.4661281.12t5.a.a$2$ $ 17^{2} \cdot 127^{2}$ 12.12.8025462320079492591473139857.1 $C_3 : C_4$ (as 12T5) $-1$ $2$
* 2.2159.6t5.a.a$2$ $ 17 \cdot 127 $ 6.6.79241777.1 $S_3\times C_3$ (as 6T5) $0$ $2$
* 2.36703.12t19.a.a$2$ $ 17^{2} \cdot 127 $ 12.12.30850000558264402577.1 $C_3\times (C_3 : C_4)$ (as 12T19) $0$ $2$
* 2.2159.6t5.a.b$2$ $ 17 \cdot 127 $ 6.6.79241777.1 $S_3\times C_3$ (as 6T5) $0$ $2$
* 2.36703.12t19.a.b$2$ $ 17^{2} \cdot 127 $ 12.12.30850000558264402577.1 $C_3\times (C_3 : C_4)$ (as 12T19) $0$ $2$

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 *.