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Properties

Label 12.0.33171021564453125.1
Degree $12$
Signature $[0, 6]$
Discriminant $3.317\times 10^{16}$
Root discriminant \(23.81\)
Ramified primes $5,19$
Class number $13$
Class group [13]
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 + 7*x^10 - 6*x^9 + 41*x^8 - 62*x^7 + 266*x^6 - 351*x^5 + 1513*x^4 - 1757*x^3 + 2107*x^2 - 2058*x + 2401)
 
gp: K = bnfinit(y^12 - y^11 + 7*y^10 - 6*y^9 + 41*y^8 - 62*y^7 + 266*y^6 - 351*y^5 + 1513*y^4 - 1757*y^3 + 2107*y^2 - 2058*y + 2401, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 + 7*x^10 - 6*x^9 + 41*x^8 - 62*x^7 + 266*x^6 - 351*x^5 + 1513*x^4 - 1757*x^3 + 2107*x^2 - 2058*x + 2401);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - x^11 + 7*x^10 - 6*x^9 + 41*x^8 - 62*x^7 + 266*x^6 - 351*x^5 + 1513*x^4 - 1757*x^3 + 2107*x^2 - 2058*x + 2401)
 

\( x^{12} - x^{11} + 7 x^{10} - 6 x^{9} + 41 x^{8} - 62 x^{7} + 266 x^{6} - 351 x^{5} + 1513 x^{4} + \cdots + 2401 \) 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:  $[0, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(33171021564453125\) \(\medspace = 5^{9}\cdot 19^{8}\) 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:  \(23.81\)
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:  $5^{3/4}19^{2/3}\approx 23.8083831956814$
Ramified primes:   \(5\), \(19\) 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:  \(95=5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{95}(64,·)$, $\chi_{95}(1,·)$, $\chi_{95}(68,·)$, $\chi_{95}(39,·)$, $\chi_{95}(11,·)$, $\chi_{95}(77,·)$, $\chi_{95}(49,·)$, $\chi_{95}(83,·)$, $\chi_{95}(87,·)$, $\chi_{95}(7,·)$, $\chi_{95}(26,·)$, $\chi_{95}(58,·)$$\rbrace$
This is a CM field.
Reflex fields:  \(\Q(\zeta_{5})\)$^{2}$, 12.0.33171021564453125.1$^{30}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{360437}a^{9}+\frac{60066}{360437}a^{8}+\frac{211}{51491}a^{7}-\frac{1511}{360437}a^{6}+\frac{70398}{360437}a^{5}-\frac{60325}{360437}a^{4}-\frac{8540}{51491}a^{3}-\frac{20595}{360437}a^{2}-\frac{39486}{360437}a-\frac{20301}{51491}$, $\frac{1}{2523059}a^{10}-\frac{1}{2523059}a^{9}-\frac{52716}{360437}a^{8}+\frac{307469}{2523059}a^{7}+\frac{1511}{2523059}a^{6}-\frac{10107}{2523059}a^{5}-\frac{50229}{360437}a^{4}+\frac{111271}{2523059}a^{3}+\frac{1101906}{2523059}a^{2}-\frac{53206}{360437}a+\frac{8828}{51491}$, $\frac{1}{17661413}a^{11}-\frac{1}{17661413}a^{10}+\frac{1}{2523059}a^{9}-\frac{6839769}{17661413}a^{8}+\frac{2584889}{17661413}a^{7}-\frac{1777}{17661413}a^{6}+\frac{422222}{2523059}a^{5}-\frac{2533406}{17661413}a^{4}+\frac{2773982}{17661413}a^{3}+\frac{236860}{2523059}a^{2}-\frac{51204}{360437}a-\frac{10052}{51491}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

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

Class group and class number

$C_{13}$, which has order $13$

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:  $5$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{41}{360437} a^{11} - \frac{33}{32767} a^{6} + \frac{294302}{360437} a \)  (order $10$) 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:   $\frac{83}{2523059}a^{11}+\frac{204}{360437}a^{10}-\frac{1224}{2523059}a^{9}+\frac{8364}{2523059}a^{8}-\frac{809}{360437}a^{7}+\frac{8115}{360437}a^{6}-\frac{71604}{2523059}a^{5}+\frac{308652}{2523059}a^{4}-\frac{51204}{360437}a^{3}+\frac{2140312}{2523059}a^{2}-\frac{354278}{360437}a+\frac{9996}{51491}$, $\frac{82}{360437}a^{11}+\frac{66}{32767}a^{6}-\frac{228167}{360437}a$, $\frac{2092}{17661413}a^{11}-\frac{2092}{17661413}a^{10}+\frac{83}{2523059}a^{9}-\frac{12552}{17661413}a^{8}+\frac{85772}{17661413}a^{7}-\frac{4184}{569723}a^{6}+\frac{79496}{2523059}a^{5}-\frac{858801}{17661413}a^{4}+\frac{3165196}{17661413}a^{3}-\frac{525092}{2523059}a^{2}+\frac{89956}{360437}a-\frac{12552}{51491}$, $\frac{5553}{2523059}a^{11}+\frac{3029}{2523059}a^{10}+\frac{5100}{360437}a^{9}+\frac{13505}{2523059}a^{8}+\frac{208080}{2523059}a^{7}-\frac{30182}{2523059}a^{6}+\frac{171861}{360437}a^{5}-\frac{248880}{2523059}a^{4}+\frac{6386361}{2523059}a^{3}+\frac{1020}{51491}a^{2}+\frac{79778}{32767}a-\frac{89849}{51491}$, $\frac{249}{2523059}a^{11}-\frac{326}{360437}a^{10}+\frac{3174}{2523059}a^{9}-\frac{21689}{2523059}a^{8}+\frac{331}{360437}a^{7}-\frac{11665}{360437}a^{6}+\frac{75184}{2523059}a^{5}-\frac{800377}{2523059}a^{4}+\frac{132779}{360437}a^{3}-\frac{3373366}{2523059}a^{2}-\frac{32072}{360437}a-\frac{98451}{51491}$ 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:  \( 1234.55163261 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1234.55163261 \cdot 13}{10\cdot\sqrt{33171021564453125}}\cr\approx \mathstrut & 0.542191116043 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 7*x^10 - 6*x^9 + 41*x^8 - 62*x^7 + 266*x^6 - 351*x^5 + 1513*x^4 - 1757*x^3 + 2107*x^2 - 2058*x + 2401)
 
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 + 7*x^10 - 6*x^9 + 41*x^8 - 62*x^7 + 266*x^6 - 351*x^5 + 1513*x^4 - 1757*x^3 + 2107*x^2 - 2058*x + 2401, 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 + 7*x^10 - 6*x^9 + 41*x^8 - 62*x^7 + 266*x^6 - 351*x^5 + 1513*x^4 - 1757*x^3 + 2107*x^2 - 2058*x + 2401);
 
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 + 7*x^10 - 6*x^9 + 41*x^8 - 62*x^7 + 266*x^6 - 351*x^5 + 1513*x^4 - 1757*x^3 + 2107*x^2 - 2058*x + 2401);
 
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

\(\Q(\sqrt{5}) \), 3.3.361.1, \(\Q(\zeta_{5})\), 6.6.16290125.1

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 ${\href{/padicField/7.4.0.1}{4} }^{3}$ ${\href{/padicField/11.1.0.1}{1} }^{12}$ ${\href{/padicField/13.12.0.1}{12} }$ ${\href{/padicField/17.12.0.1}{12} }$ R ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.1.0.1}{1} }^{12}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.3.0.1}{3} }^{4}$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.12.0.1}{12} }$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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
\(5\) 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}$
\(19\) Copy content Toggle raw display 19.6.4.3$x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.6.4.3$x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.5.4t1.a.a$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 1.5.4t1.a.b$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
* 1.19.3t1.a.a$1$ $ 19 $ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
* 1.95.12t1.a.a$1$ $ 5 \cdot 19 $ 12.0.33171021564453125.1 $C_{12}$ (as 12T1) $0$ $-1$
* 1.95.6t1.a.a$1$ $ 5 \cdot 19 $ 6.6.16290125.1 $C_6$ (as 6T1) $0$ $1$
* 1.95.12t1.a.b$1$ $ 5 \cdot 19 $ 12.0.33171021564453125.1 $C_{12}$ (as 12T1) $0$ $-1$
* 1.19.3t1.a.b$1$ $ 19 $ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
* 1.95.12t1.a.c$1$ $ 5 \cdot 19 $ 12.0.33171021564453125.1 $C_{12}$ (as 12T1) $0$ $-1$
* 1.95.6t1.a.b$1$ $ 5 \cdot 19 $ 6.6.16290125.1 $C_6$ (as 6T1) $0$ $1$
* 1.95.12t1.a.d$1$ $ 5 \cdot 19 $ 12.0.33171021564453125.1 $C_{12}$ (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 *.