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Properties

Label 12.12.187...125.1
Degree $12$
Signature $[12, 0]$
Discriminant $1.874\times 10^{20}$
Root discriminant \(48.91\)
Ramified primes $5,7,13$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 45*x^10 + 15*x^9 + 661*x^8 - 70*x^7 - 3836*x^6 + 660*x^5 + 7876*x^4 - 3890*x^3 - 1930*x^2 + 1489*x - 229)
 
gp: K = bnfinit(y^12 - y^11 - 45*y^10 + 15*y^9 + 661*y^8 - 70*y^7 - 3836*y^6 + 660*y^5 + 7876*y^4 - 3890*y^3 - 1930*y^2 + 1489*y - 229, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 45*x^10 + 15*x^9 + 661*x^8 - 70*x^7 - 3836*x^6 + 660*x^5 + 7876*x^4 - 3890*x^3 - 1930*x^2 + 1489*x - 229);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - x^11 - 45*x^10 + 15*x^9 + 661*x^8 - 70*x^7 - 3836*x^6 + 660*x^5 + 7876*x^4 - 3890*x^3 - 1930*x^2 + 1489*x - 229)
 

\( x^{12} - x^{11} - 45 x^{10} + 15 x^{9} + 661 x^{8} - 70 x^{7} - 3836 x^{6} + 660 x^{5} + 7876 x^{4} + \cdots - 229 \) 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:   \(187441217958845703125\) \(\medspace = 5^{9}\cdot 7^{6}\cdot 13^{8}\) 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:  \(48.91\)
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:  $5^{3/4}7^{1/2}13^{2/3}\approx 48.91087415715005$
Ramified primes:   \(5\), \(7\), \(13\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(455=5\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{455}(1,·)$, $\chi_{455}(386,·)$, $\chi_{455}(328,·)$, $\chi_{455}(204,·)$, $\chi_{455}(237,·)$, $\chi_{455}(48,·)$, $\chi_{455}(274,·)$, $\chi_{455}(211,·)$, $\chi_{455}(118,·)$, $\chi_{455}(27,·)$, $\chi_{455}(412,·)$, $\chi_{455}(29,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{279400884510389}a^{11}+\frac{29061878898295}{279400884510389}a^{10}+\frac{65462489360028}{279400884510389}a^{9}-\frac{107101996124666}{279400884510389}a^{8}+\frac{61432933545493}{279400884510389}a^{7}-\frac{19336941590571}{279400884510389}a^{6}+\frac{117966429061760}{279400884510389}a^{5}-\frac{117042507713111}{279400884510389}a^{4}-\frac{31545777749250}{279400884510389}a^{3}+\frac{20030515904690}{279400884510389}a^{2}-\frac{22356405171257}{279400884510389}a-\frac{128748010987083}{279400884510389}$ 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 $1$ (assuming GRH)

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:   $\frac{119705967}{1225501601}a^{11}-\frac{65238188}{1225501601}a^{10}-\frac{5419656630}{1225501601}a^{9}-\frac{659327925}{1225501601}a^{8}+\frac{78939357148}{1225501601}a^{7}+\frac{27221468564}{1225501601}a^{6}-\frac{448053082431}{1225501601}a^{5}-\frac{121817865120}{1225501601}a^{4}+\frac{891572781389}{1225501601}a^{3}-\frac{70230906547}{1225501601}a^{2}-\frac{263067817935}{1225501601}a+\frac{62630943626}{1225501601}$, $\frac{11627722850145}{279400884510389}a^{11}-\frac{6165425055554}{279400884510389}a^{10}-\frac{525968562708610}{279400884510389}a^{9}-\frac{72334998519795}{279400884510389}a^{8}+\frac{76\!\cdots\!50}{279400884510389}a^{7}+\frac{27\!\cdots\!90}{279400884510389}a^{6}-\frac{43\!\cdots\!19}{279400884510389}a^{5}-\frac{12\!\cdots\!15}{279400884510389}a^{4}+\frac{84\!\cdots\!95}{279400884510389}a^{3}-\frac{57\!\cdots\!65}{279400884510389}a^{2}-\frac{23\!\cdots\!90}{279400884510389}a+\frac{55\!\cdots\!86}{279400884510389}$, $\frac{29921766317970}{279400884510389}a^{11}-\frac{15730205334300}{279400884510389}a^{10}-\frac{13\!\cdots\!40}{279400884510389}a^{9}-\frac{193445099490445}{279400884510389}a^{8}+\frac{19\!\cdots\!20}{279400884510389}a^{7}+\frac{72\!\cdots\!10}{279400884510389}a^{6}-\frac{11\!\cdots\!24}{279400884510389}a^{5}-\frac{33\!\cdots\!95}{279400884510389}a^{4}+\frac{22\!\cdots\!00}{279400884510389}a^{3}-\frac{11\!\cdots\!25}{279400884510389}a^{2}-\frac{64\!\cdots\!15}{279400884510389}a+\frac{13\!\cdots\!34}{279400884510389}$, $\frac{11763890018420}{279400884510389}a^{11}-\frac{7006458261541}{279400884510389}a^{10}-\frac{533260657404574}{279400884510389}a^{9}-\frac{36108750556993}{279400884510389}a^{8}+\frac{78\!\cdots\!69}{279400884510389}a^{7}+\frac{22\!\cdots\!96}{279400884510389}a^{6}-\frac{44\!\cdots\!05}{279400884510389}a^{5}-\frac{93\!\cdots\!27}{279400884510389}a^{4}+\frac{90\!\cdots\!15}{279400884510389}a^{3}-\frac{12\!\cdots\!59}{279400884510389}a^{2}-\frac{26\!\cdots\!71}{279400884510389}a+\frac{73\!\cdots\!16}{279400884510389}$, $\frac{15006158192096}{279400884510389}a^{11}-\frac{6055593922061}{279400884510389}a^{10}-\frac{680400589312170}{279400884510389}a^{9}-\frac{176154022648984}{279400884510389}a^{8}+\frac{98\!\cdots\!42}{279400884510389}a^{7}+\frac{47\!\cdots\!03}{279400884510389}a^{6}-\frac{55\!\cdots\!77}{279400884510389}a^{5}-\frac{22\!\cdots\!33}{279400884510389}a^{4}+\frac{10\!\cdots\!32}{279400884510389}a^{3}+\frac{29\!\cdots\!69}{279400884510389}a^{2}-\frac{29\!\cdots\!99}{279400884510389}a+\frac{51\!\cdots\!28}{279400884510389}$, $\frac{11495153901981}{279400884510389}a^{11}-\frac{7485291926690}{279400884510389}a^{10}-\frac{521495898865240}{279400884510389}a^{9}-\frac{4029094429065}{279400884510389}a^{8}+\frac{76\!\cdots\!74}{279400884510389}a^{7}+\frac{17\!\cdots\!74}{279400884510389}a^{6}-\frac{44\!\cdots\!41}{279400884510389}a^{5}-\frac{64\!\cdots\!95}{279400884510389}a^{4}+\frac{90\!\cdots\!97}{279400884510389}a^{3}-\frac{17\!\cdots\!92}{279400884510389}a^{2}-\frac{27\!\cdots\!02}{279400884510389}a+\frac{87\!\cdots\!34}{279400884510389}$, $\frac{57213410028333}{279400884510389}a^{11}-\frac{30603794578232}{279400884510389}a^{10}-\frac{25\!\cdots\!10}{279400884510389}a^{9}-\frac{343764613783270}{279400884510389}a^{8}+\frac{37\!\cdots\!92}{279400884510389}a^{7}+\frac{13\!\cdots\!06}{279400884510389}a^{6}-\frac{21\!\cdots\!83}{279400884510389}a^{5}-\frac{61\!\cdots\!75}{279400884510389}a^{4}+\frac{42\!\cdots\!21}{279400884510389}a^{3}-\frac{27\!\cdots\!08}{279400884510389}a^{2}-\frac{12\!\cdots\!41}{279400884510389}a+\frac{27\!\cdots\!59}{279400884510389}$, $\frac{48705770641863}{279400884510389}a^{11}-\frac{26807476494069}{279400884510389}a^{10}-\frac{22\!\cdots\!41}{279400884510389}a^{9}-\frac{263069703557313}{279400884510389}a^{8}+\frac{32\!\cdots\!89}{279400884510389}a^{7}+\frac{11\!\cdots\!44}{279400884510389}a^{6}-\frac{18\!\cdots\!90}{279400884510389}a^{5}-\frac{50\!\cdots\!85}{279400884510389}a^{4}+\frac{35\!\cdots\!70}{279400884510389}a^{3}-\frac{25\!\cdots\!46}{279400884510389}a^{2}-\frac{10\!\cdots\!09}{279400884510389}a+\frac{24\!\cdots\!65}{279400884510389}$, $\frac{69978581433549}{279400884510389}a^{11}-\frac{37080906866813}{279400884510389}a^{10}-\frac{31\!\cdots\!22}{279400884510389}a^{9}-\frac{431742865679577}{279400884510389}a^{8}+\frac{46\!\cdots\!63}{279400884510389}a^{7}+\frac{16\!\cdots\!81}{279400884510389}a^{6}-\frac{26\!\cdots\!99}{279400884510389}a^{5}-\frac{74\!\cdots\!93}{279400884510389}a^{4}+\frac{51\!\cdots\!17}{279400884510389}a^{3}-\frac{34\!\cdots\!30}{279400884510389}a^{2}-\frac{15\!\cdots\!78}{279400884510389}a+\frac{35\!\cdots\!66}{279400884510389}$, $\frac{332440631996209}{279400884510389}a^{11}-\frac{185305454970950}{279400884510389}a^{10}-\frac{15\!\cdots\!61}{279400884510389}a^{9}-\frac{16\!\cdots\!86}{279400884510389}a^{8}+\frac{21\!\cdots\!61}{279400884510389}a^{7}+\frac{73\!\cdots\!29}{279400884510389}a^{6}-\frac{12\!\cdots\!67}{279400884510389}a^{5}-\frac{33\!\cdots\!06}{279400884510389}a^{4}+\frac{24\!\cdots\!94}{279400884510389}a^{3}-\frac{19\!\cdots\!87}{279400884510389}a^{2}-\frac{72\!\cdots\!51}{279400884510389}a+\frac{17\!\cdots\!31}{279400884510389}$, $\frac{37200773399435}{279400884510389}a^{11}-\frac{21618664724476}{279400884510389}a^{10}-\frac{16\!\cdots\!86}{279400884510389}a^{9}-\frac{154850155096207}{279400884510389}a^{8}+\frac{24\!\cdots\!38}{279400884510389}a^{7}+\frac{79\!\cdots\!37}{279400884510389}a^{6}-\frac{13\!\cdots\!74}{279400884510389}a^{5}-\frac{36\!\cdots\!54}{279400884510389}a^{4}+\frac{27\!\cdots\!38}{279400884510389}a^{3}-\frac{18\!\cdots\!81}{279400884510389}a^{2}-\frac{79\!\cdots\!22}{279400884510389}a+\frac{18\!\cdots\!09}{279400884510389}$ Copy content Toggle raw display (assuming GRH)
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:  \( 1464478.29313 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 1464478.29313 \cdot 1}{2\cdot\sqrt{187441217958845703125}}\cr\approx \mathstrut & 0.219068705702 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 45*x^10 + 15*x^9 + 661*x^8 - 70*x^7 - 3836*x^6 + 660*x^5 + 7876*x^4 - 3890*x^3 - 1930*x^2 + 1489*x - 229)
 
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 - 45*x^10 + 15*x^9 + 661*x^8 - 70*x^7 - 3836*x^6 + 660*x^5 + 7876*x^4 - 3890*x^3 - 1930*x^2 + 1489*x - 229, 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 - 45*x^10 + 15*x^9 + 661*x^8 - 70*x^7 - 3836*x^6 + 660*x^5 + 7876*x^4 - 3890*x^3 - 1930*x^2 + 1489*x - 229);
 
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 - 45*x^10 + 15*x^9 + 661*x^8 - 70*x^7 - 3836*x^6 + 660*x^5 + 7876*x^4 - 3890*x^3 - 1930*x^2 + 1489*x - 229);
 
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

$C_{12}$ (as 12T1):

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 cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.169.1, 4.4.6125.1, 6.6.3570125.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.12.0.1}{12} }$ ${\href{/padicField/3.12.0.1}{12} }$ R R ${\href{/padicField/11.3.0.1}{3} }^{4}$ R ${\href{/padicField/17.12.0.1}{12} }$ ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.4.0.1}{4} }^{3}$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.3.0.1}{3} }^{4}$

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
\(5\) Copy content Toggle raw display 5.4.3.1$x^{4} + 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} + 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} + 20$$4$$1$$3$$C_4$$[\ ]_{4}$
\(7\) Copy content Toggle raw display 7.12.6.2$x^{12} + 49 x^{8} - 1715 x^{6} + 9604 x^{4} - 100842 x^{2} + 352947$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
\(13\) Copy content Toggle raw display 13.12.8.1$x^{12} + 9 x^{10} + 88 x^{9} + 33 x^{8} + 216 x^{7} - 1299 x^{6} - 78 x^{5} - 1797 x^{4} - 15494 x^{3} + 21687 x^{2} - 41586 x + 201846$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.35.4t1.a.a$1$ $ 5 \cdot 7 $ 4.4.6125.1 $C_4$ (as 4T1) $0$ $1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 1.35.4t1.a.b$1$ $ 5 \cdot 7 $ 4.4.6125.1 $C_4$ (as 4T1) $0$ $1$
* 1.13.3t1.a.a$1$ $ 13 $ 3.3.169.1 $C_3$ (as 3T1) $0$ $1$
* 1.455.12t1.a.a$1$ $ 5 \cdot 7 \cdot 13 $ 12.12.187441217958845703125.1 $C_{12}$ (as 12T1) $0$ $1$
* 1.65.6t1.b.a$1$ $ 5 \cdot 13 $ 6.6.3570125.1 $C_6$ (as 6T1) $0$ $1$
* 1.455.12t1.a.b$1$ $ 5 \cdot 7 \cdot 13 $ 12.12.187441217958845703125.1 $C_{12}$ (as 12T1) $0$ $1$
* 1.13.3t1.a.b$1$ $ 13 $ 3.3.169.1 $C_3$ (as 3T1) $0$ $1$
* 1.455.12t1.a.c$1$ $ 5 \cdot 7 \cdot 13 $ 12.12.187441217958845703125.1 $C_{12}$ (as 12T1) $0$ $1$
* 1.65.6t1.b.b$1$ $ 5 \cdot 13 $ 6.6.3570125.1 $C_6$ (as 6T1) $0$ $1$
* 1.455.12t1.a.d$1$ $ 5 \cdot 7 \cdot 13 $ 12.12.187441217958845703125.1 $C_{12}$ (as 12T1) $0$ $1$

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 *.