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Properties

Label 12.12.46118408000000000.1
Degree $12$
Signature $[12, 0]$
Discriminant $4.612\times 10^{16}$
Root discriminant \(24.47\)
Ramified primes $2,5,7$
Class number $1$
Class group trivial
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 - 4*x^11 - 17*x^10 + 74*x^9 + 74*x^8 - 412*x^7 - 23*x^6 + 734*x^5 - 175*x^4 - 324*x^3 + 90*x^2 + 22*x + 1)
 
gp: K = bnfinit(y^12 - 4*y^11 - 17*y^10 + 74*y^9 + 74*y^8 - 412*y^7 - 23*y^6 + 734*y^5 - 175*y^4 - 324*y^3 + 90*y^2 + 22*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - 17*x^10 + 74*x^9 + 74*x^8 - 412*x^7 - 23*x^6 + 734*x^5 - 175*x^4 - 324*x^3 + 90*x^2 + 22*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 - 17*x^10 + 74*x^9 + 74*x^8 - 412*x^7 - 23*x^6 + 734*x^5 - 175*x^4 - 324*x^3 + 90*x^2 + 22*x + 1)
 

\( x^{12} - 4 x^{11} - 17 x^{10} + 74 x^{9} + 74 x^{8} - 412 x^{7} - 23 x^{6} + 734 x^{5} - 175 x^{4} + \cdots + 1 \) 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:   \(46118408000000000\) \(\medspace = 2^{12}\cdot 5^{9}\cdot 7^{8}\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(140=2^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{140}(1,·)$, $\chi_{140}(67,·)$, $\chi_{140}(9,·)$, $\chi_{140}(107,·)$, $\chi_{140}(109,·)$, $\chi_{140}(81,·)$, $\chi_{140}(43,·)$, $\chi_{140}(23,·)$, $\chi_{140}(121,·)$, $\chi_{140}(123,·)$, $\chi_{140}(29,·)$, $\chi_{140}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{239}a^{10}-\frac{44}{239}a^{9}+\frac{74}{239}a^{8}+\frac{45}{239}a^{7}+\frac{4}{239}a^{6}+\frac{86}{239}a^{5}-\frac{101}{239}a^{4}+\frac{99}{239}a^{3}+\frac{2}{239}a^{2}-\frac{8}{239}a-\frac{60}{239}$, $\frac{1}{6715661}a^{11}+\frac{7910}{6715661}a^{10}+\frac{1933982}{6715661}a^{9}-\frac{2998271}{6715661}a^{8}+\frac{1626068}{6715661}a^{7}-\frac{1904715}{6715661}a^{6}+\frac{1845244}{6715661}a^{5}-\frac{2486054}{6715661}a^{4}-\frac{1903453}{6715661}a^{3}-\frac{2807168}{6715661}a^{2}-\frac{970458}{6715661}a-\frac{2405731}{6715661}$ 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:   $\frac{28}{239}a^{11}-\frac{180}{239}a^{10}-\frac{240}{239}a^{9}+\frac{3345}{239}a^{8}-\frac{2260}{239}a^{7}-\frac{18718}{239}a^{6}+\frac{23114}{239}a^{5}+\frac{33945}{239}a^{4}-190a^{3}-\frac{17479}{239}a^{2}+\frac{16672}{239}a+\frac{1410}{239}$, $\frac{3437580}{6715661}a^{11}-\frac{12901446}{6715661}a^{10}-\frac{61192460}{6715661}a^{9}+\frac{237806895}{6715661}a^{8}+\frac{304862760}{6715661}a^{7}-\frac{1314226260}{6715661}a^{6}-\frac{354933090}{6715661}a^{5}+\frac{2289625920}{6715661}a^{4}-\frac{139666710}{6715661}a^{3}-\frac{901616625}{6715661}a^{2}+\frac{171192220}{6715661}a+\frac{28812587}{6715661}$, $\frac{3437580}{6715661}a^{11}-\frac{12901446}{6715661}a^{10}-\frac{61192460}{6715661}a^{9}+\frac{237806895}{6715661}a^{8}+\frac{304862760}{6715661}a^{7}-\frac{1314226260}{6715661}a^{6}-\frac{354933090}{6715661}a^{5}+\frac{2289625920}{6715661}a^{4}-\frac{139666710}{6715661}a^{3}-\frac{901616625}{6715661}a^{2}+\frac{171192220}{6715661}a+\frac{22096926}{6715661}$, $\frac{9043088}{6715661}a^{11}-\frac{31285635}{6715661}a^{10}-\frac{170094410}{6715661}a^{9}+\frac{575365225}{6715661}a^{8}+\frac{970042164}{6715661}a^{7}-\frac{3166608661}{6715661}a^{6}-\frac{1863872584}{6715661}a^{5}+\frac{5439387375}{6715661}a^{4}+\frac{1258811550}{6715661}a^{3}-\frac{1927499249}{6715661}a^{2}-\frac{166269830}{6715661}a+\frac{613570}{6715661}$, $\frac{2795684}{6715661}a^{11}-\frac{10758279}{6715661}a^{10}-\frac{48857624}{6715661}a^{9}+\frac{198566871}{6715661}a^{8}+\frac{230995422}{6715661}a^{7}-\frac{1100192562}{6715661}a^{6}-\frac{193769364}{6715661}a^{5}+\frac{1928913675}{6715661}a^{4}-\frac{276511906}{6715661}a^{3}-\frac{771688131}{6715661}a^{2}+\frac{179873706}{6715661}a+\frac{22945800}{6715661}$, $\frac{1195880}{6715661}a^{11}-\frac{2365470}{6715661}a^{10}-\frac{28776767}{6715661}a^{9}+\frac{43338198}{6715661}a^{8}+\frac{243571048}{6715661}a^{7}-\frac{238795361}{6715661}a^{6}-\frac{877952033}{6715661}a^{5}+\frac{404649810}{6715661}a^{4}+\frac{1239159111}{6715661}a^{3}-\frac{77423612}{6715661}a^{2}-\frac{392017617}{6715661}a-\frac{25251942}{6715661}$, $\frac{1454928}{6715661}a^{11}-\frac{5478156}{6715661}a^{10}-\frac{25671933}{6715661}a^{9}+\frac{100477542}{6715661}a^{8}+\frac{124795452}{6715661}a^{7}-\frac{549473817}{6715661}a^{6}-\frac{126461343}{6715661}a^{5}+\frac{931155555}{6715661}a^{4}-\frac{102850231}{6715661}a^{3}-\frac{333050592}{6715661}a^{2}+\frac{94743309}{6715661}a+\frac{7729278}{6715661}$, $\frac{4555930}{6715661}a^{11}-\frac{13863591}{6715661}a^{10}-\frac{92351193}{6715661}a^{9}+\frac{254427522}{6715661}a^{8}+\frac{611072136}{6715661}a^{7}-\frac{1395946623}{6715661}a^{6}-\frac{1611442818}{6715661}a^{5}+\frac{2367846420}{6715661}a^{4}+\frac{1783842992}{6715661}a^{3}-\frac{724886480}{6715661}a^{2}-\frac{486854428}{6715661}a-\frac{38704793}{6715661}$, $\frac{7016888}{6715661}a^{11}-\frac{24367166}{6715661}a^{10}-\frac{131651320}{6715661}a^{9}+\frac{448250255}{6715661}a^{8}+\frac{746196260}{6715661}a^{7}-\frac{2468484088}{6715661}a^{6}-\frac{1406093380}{6715661}a^{5}+\frac{4248485740}{6715661}a^{4}+\frac{889765000}{6715661}a^{3}-\frac{1522527564}{6715661}a^{2}-\frac{73234767}{6715661}a+\frac{6489606}{6715661}$, $a$, $\frac{9403464}{6715661}a^{11}-\frac{37817795}{6715661}a^{10}-\frac{158475937}{6715661}a^{9}+\frac{697470083}{6715661}a^{8}+\frac{670116894}{6715661}a^{7}-\frac{3855838371}{6715661}a^{6}-\frac{72430425}{6715661}a^{5}+\frac{6726570160}{6715661}a^{4}-\frac{1901881607}{6715661}a^{3}-\frac{2707934504}{6715661}a^{2}+\frac{967123084}{6715661}a+\frac{80875616}{6715661}$ 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:  \( 17863.7216242 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 17863.7216242 \cdot 1}{2\cdot\sqrt{46118408000000000}}\cr\approx \mathstrut & 0.170358866239 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 17*x^10 + 74*x^9 + 74*x^8 - 412*x^7 - 23*x^6 + 734*x^5 - 175*x^4 - 324*x^3 + 90*x^2 + 22*x + 1)
 
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 - 4*x^11 - 17*x^10 + 74*x^9 + 74*x^8 - 412*x^7 - 23*x^6 + 734*x^5 - 175*x^4 - 324*x^3 + 90*x^2 + 22*x + 1, 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 - 4*x^11 - 17*x^10 + 74*x^9 + 74*x^8 - 412*x^7 - 23*x^6 + 734*x^5 - 175*x^4 - 324*x^3 + 90*x^2 + 22*x + 1);
 
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 - 4*x^11 - 17*x^10 + 74*x^9 + 74*x^8 - 412*x^7 - 23*x^6 + 734*x^5 - 175*x^4 - 324*x^3 + 90*x^2 + 22*x + 1);
 
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{20})^+\), 6.6.300125.1

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 R ${\href{/padicField/3.12.0.1}{12} }$ R R ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{3}$ ${\href{/padicField/17.12.0.1}{12} }$ ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.2.0.1}{2} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.1.0.1}{1} }^{12}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.12.0.1}{12} }$ ${\href{/padicField/59.3.0.1}{3} }^{4}$

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
\(2\) Copy content Toggle raw display 2.12.12.25$x^{12} + 12 x^{11} + 60 x^{10} + 160 x^{9} + 308 x^{8} + 736 x^{7} + 2272 x^{6} + 5632 x^{5} + 10608 x^{4} + 15040 x^{3} + 12224 x^{2} + 3584 x + 704$$2$$6$$12$$C_{12}$$[2]^{6}$
\(5\) Copy content Toggle raw display 5.12.9.2$x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
\(7\) 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}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.20.4t1.a.a$1$ $ 2^{2} \cdot 5 $ \(\Q(\zeta_{20})^+\) $C_4$ (as 4T1) $0$ $1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 1.20.4t1.a.b$1$ $ 2^{2} \cdot 5 $ \(\Q(\zeta_{20})^+\) $C_4$ (as 4T1) $0$ $1$
* 1.7.3t1.a.a$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
* 1.140.12t1.a.a$1$ $ 2^{2} \cdot 5 \cdot 7 $ 12.12.46118408000000000.1 $C_{12}$ (as 12T1) $0$ $1$
* 1.35.6t1.b.a$1$ $ 5 \cdot 7 $ 6.6.300125.1 $C_6$ (as 6T1) $0$ $1$
* 1.140.12t1.a.b$1$ $ 2^{2} \cdot 5 \cdot 7 $ 12.12.46118408000000000.1 $C_{12}$ (as 12T1) $0$ $1$
* 1.7.3t1.a.b$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
* 1.140.12t1.a.c$1$ $ 2^{2} \cdot 5 \cdot 7 $ 12.12.46118408000000000.1 $C_{12}$ (as 12T1) $0$ $1$
* 1.35.6t1.b.b$1$ $ 5 \cdot 7 $ 6.6.300125.1 $C_6$ (as 6T1) $0$ $1$
* 1.140.12t1.a.d$1$ $ 2^{2} \cdot 5 \cdot 7 $ 12.12.46118408000000000.1 $C_{12}$ (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 *.