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Properties

Label 12.12.683...097.1
Degree $12$
Signature $[12, 0]$
Discriminant $6.836\times 10^{17}$
Root discriminant \(30.64\)
Ramified primes $7,17$
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 - 28*x^10 + 31*x^9 + 232*x^8 - 249*x^7 - 742*x^6 + 716*x^5 + 925*x^4 - 785*x^3 - 388*x^2 + 288*x + 13)
 
gp: K = bnfinit(y^12 - y^11 - 28*y^10 + 31*y^9 + 232*y^8 - 249*y^7 - 742*y^6 + 716*y^5 + 925*y^4 - 785*y^3 - 388*y^2 + 288*y + 13, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 28*x^10 + 31*x^9 + 232*x^8 - 249*x^7 - 742*x^6 + 716*x^5 + 925*x^4 - 785*x^3 - 388*x^2 + 288*x + 13);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 28*x^10 + 31*x^9 + 232*x^8 - 249*x^7 - 742*x^6 + 716*x^5 + 925*x^4 - 785*x^3 - 388*x^2 + 288*x + 13)
 

\( x^{12} - x^{11} - 28 x^{10} + 31 x^{9} + 232 x^{8} - 249 x^{7} - 742 x^{6} + 716 x^{5} + 925 x^{4} + \cdots + 13 \) 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:   \(683635509017782097\) \(\medspace = 7^{8}\cdot 17^{9}\) 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:  \(30.64\)
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:  $7^{2/3}17^{3/4}\approx 30.636234448963954$
Ramified primes:   \(7\), \(17\) 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:  \(\Q(\sqrt{17}) \)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(119=7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{119}(64,·)$, $\chi_{119}(1,·)$, $\chi_{119}(67,·)$, $\chi_{119}(4,·)$, $\chi_{119}(72,·)$, $\chi_{119}(106,·)$, $\chi_{119}(18,·)$, $\chi_{119}(16,·)$, $\chi_{119}(81,·)$, $\chi_{119}(50,·)$, $\chi_{119}(86,·)$, $\chi_{119}(30,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{26}a^{10}+\frac{5}{26}a^{9}+\frac{3}{26}a^{8}-\frac{11}{26}a^{7}-\frac{1}{2}a^{5}-\frac{1}{26}a^{4}-\frac{5}{26}a^{3}-\frac{3}{26}a^{2}+\frac{11}{26}a$, $\frac{1}{1257508226}a^{11}-\frac{223763}{96731402}a^{10}-\frac{14378841}{1257508226}a^{9}+\frac{1362805}{96731402}a^{8}-\frac{29079848}{628754113}a^{7}+\frac{23510455}{96731402}a^{6}+\frac{447519435}{1257508226}a^{5}-\frac{23296539}{96731402}a^{4}-\frac{282073237}{1257508226}a^{3}-\frac{27729885}{96731402}a^{2}-\frac{31149152}{628754113}a-\frac{513474}{48365701}$ 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:   $\frac{194301961}{628754113}a^{11}-\frac{22811813}{628754113}a^{10}-\frac{5488778574}{628754113}a^{9}+\frac{1213760645}{628754113}a^{8}+\frac{46902107283}{628754113}a^{7}-\frac{615693990}{48365701}a^{6}-\frac{156799562721}{628754113}a^{5}+\frac{8122460279}{628754113}a^{4}+\frac{200903468486}{628754113}a^{3}+\frac{7948625445}{628754113}a^{2}-\frac{78601234262}{628754113}a-\frac{280736624}{48365701}$, $\frac{157749015}{628754113}a^{11}-\frac{39049889}{628754113}a^{10}-\frac{4434839144}{628754113}a^{9}+\frac{1538342288}{628754113}a^{8}+\frac{37453049031}{628754113}a^{7}-\frac{819397522}{48365701}a^{6}-\frac{122928854667}{628754113}a^{5}+\frac{17065446861}{628754113}a^{4}+\frac{154007883960}{628754113}a^{3}+\frac{124227296}{628754113}a^{2}-\frac{58955720546}{628754113}a-\frac{232367843}{48365701}$, $\frac{209360035}{1257508226}a^{11}-\frac{53068963}{1257508226}a^{10}-\frac{2928947569}{628754113}a^{9}+\frac{1026597983}{628754113}a^{8}+\frac{48958849737}{1257508226}a^{7}-\frac{529801412}{48365701}a^{6}-\frac{157640187693}{1257508226}a^{5}+\frac{9871659466}{628754113}a^{4}+\frac{95449156610}{628754113}a^{3}+\frac{2188563811}{628754113}a^{2}-\frac{70371636067}{1257508226}a-\frac{512693953}{96731402}$, $\frac{194301961}{628754113}a^{11}-\frac{22811813}{628754113}a^{10}-\frac{5488778574}{628754113}a^{9}+\frac{1213760645}{628754113}a^{8}+\frac{46902107283}{628754113}a^{7}-\frac{615693990}{48365701}a^{6}-\frac{156799562721}{628754113}a^{5}+\frac{8122460279}{628754113}a^{4}+\frac{200903468486}{628754113}a^{3}+\frac{7948625445}{628754113}a^{2}-\frac{77972480149}{628754113}a-\frac{280736624}{48365701}$, $\frac{792098587}{1257508226}a^{11}-\frac{218325479}{1257508226}a^{10}-\frac{11125091867}{628754113}a^{9}+\frac{4159848967}{628754113}a^{8}+\frac{187455914001}{1257508226}a^{7}-\frac{2232289596}{48365701}a^{6}-\frac{612208496829}{1257508226}a^{5}+\frac{49542096517}{628754113}a^{4}+\frac{379510057010}{628754113}a^{3}-\frac{7859729190}{628754113}a^{2}-\frac{285973968545}{1257508226}a-\frac{1019443783}{96731402}$, $\frac{406562851}{1257508226}a^{11}+\frac{32478351}{1257508226}a^{10}-\frac{11602208965}{1257508226}a^{9}+\frac{399232237}{1257508226}a^{8}+\frac{50770229866}{628754113}a^{7}-\frac{84904055}{96731402}a^{6}-\frac{353048999595}{1257508226}a^{5}-\frac{20330535377}{1257508226}a^{4}+\frac{477680614901}{1257508226}a^{3}+\frac{36459452899}{1257508226}a^{2}-\frac{97049832092}{628754113}a-\frac{222561908}{48365701}$, $\frac{32341637}{1257508226}a^{11}+\frac{81578332}{628754113}a^{10}-\frac{82488243}{96731402}a^{9}-\frac{4336686499}{1257508226}a^{8}+\frac{6318078344}{628754113}a^{7}+\frac{1298164997}{48365701}a^{6}-\frac{31060264321}{628754113}a^{5}-\frac{94644055449}{1257508226}a^{4}+\frac{9192451009}{96731402}a^{3}+\frac{81133665875}{1257508226}a^{2}-\frac{35560910248}{628754113}a-\frac{597654279}{96731402}$, $\frac{301591937}{628754113}a^{11}-\frac{341163269}{1257508226}a^{10}-\frac{8366917265}{628754113}a^{9}+\frac{5537236015}{628754113}a^{8}+\frac{68267360693}{628754113}a^{7}-\frac{6182575039}{96731402}a^{6}-\frac{419932196665}{1257508226}a^{5}+\frac{85930968756}{628754113}a^{4}+\frac{233640682904}{628754113}a^{3}-\frac{43984711913}{628754113}a^{2}-\frac{73068290684}{628754113}a-\frac{319534667}{96731402}$, $\frac{133620261}{1257508226}a^{11}-\frac{43578369}{1257508226}a^{10}-\frac{1880652577}{628754113}a^{9}+\frac{797454348}{628754113}a^{8}+\frac{31795483101}{1257508226}a^{7}-\frac{441545331}{48365701}a^{6}-\frac{104355299901}{1257508226}a^{5}+\frac{11302495095}{628754113}a^{4}+\frac{65026508220}{628754113}a^{3}-\frac{5400637205}{628754113}a^{2}-\frac{48845445693}{1257508226}a+\frac{124090031}{96731402}$, $\frac{116772977}{1257508226}a^{11}+\frac{34461931}{628754113}a^{10}-\frac{1703245151}{628754113}a^{9}-\frac{750273715}{628754113}a^{8}+\frac{31326376317}{1257508226}a^{7}+\frac{923363681}{96731402}a^{6}-\frac{58589492206}{628754113}a^{5}-\frac{19352800373}{628754113}a^{4}+\frac{87296481776}{628754113}a^{3}+\frac{18575679599}{628754113}a^{2}-\frac{81603255879}{1257508226}a-\frac{105845481}{48365701}$, $\frac{197748905}{628754113}a^{11}-\frac{65133936}{628754113}a^{10}-\frac{11157490147}{1257508226}a^{9}+\frac{4733187711}{1257508226}a^{8}+\frac{94827350531}{1257508226}a^{7}-\frac{2587746771}{96731402}a^{6}-\frac{157648974473}{628754113}a^{5}+\frac{61904687943}{1257508226}a^{4}+\frac{404477715263}{1257508226}a^{3}-\frac{13376590609}{1257508226}a^{2}-\frac{155191563739}{1257508226}a-\frac{406288991}{96731402}$ 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:  \( 122189.376313 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 122189.376313 \cdot 1}{2\cdot\sqrt{683635509017782097}}\cr\approx \mathstrut & 0.302657258659 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 28*x^10 + 31*x^9 + 232*x^8 - 249*x^7 - 742*x^6 + 716*x^5 + 925*x^4 - 785*x^3 - 388*x^2 + 288*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 - 28*x^10 + 31*x^9 + 232*x^8 - 249*x^7 - 742*x^6 + 716*x^5 + 925*x^4 - 785*x^3 - 388*x^2 + 288*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 - 28*x^10 + 31*x^9 + 232*x^8 - 249*x^7 - 742*x^6 + 716*x^5 + 925*x^4 - 785*x^3 - 388*x^2 + 288*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 - 28*x^10 + 31*x^9 + 232*x^8 - 249*x^7 - 742*x^6 + 716*x^5 + 925*x^4 - 785*x^3 - 388*x^2 + 288*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

$C_{12}$ (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

\(\Q(\sqrt{17}) \), \(\Q(\zeta_{7})^+\), 4.4.4913.1, 6.6.11796113.1

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 ${\href{/padicField/2.6.0.1}{6} }^{2}$ ${\href{/padicField/3.12.0.1}{12} }$ ${\href{/padicField/5.12.0.1}{12} }$ R ${\href{/padicField/11.12.0.1}{12} }$ ${\href{/padicField/13.1.0.1}{1} }^{12}$ R ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.4.0.1}{4} }^{3}$ ${\href{/padicField/31.12.0.1}{12} }$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.4.0.1}{4} }^{3}$ ${\href{/padicField/43.2.0.1}{2} }^{6}$ ${\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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
\(7\) Copy content Toggle raw display 7.12.8.1$x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
\(17\) 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}$