Properties

Label 12.0.61132828589969773.1
Degree $12$
Signature $[0, 6]$
Discriminant $6.113\times 10^{16}$
Root discriminant \(25.05\)
Ramified primes $7,13$
Class number $4$
Class group [2, 2]
Galois group $C_{12}$ (as 12T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 4*x^10 - 10*x^9 + 39*x^8 - 78*x^7 + 214*x^6 - 280*x^5 + 693*x^4 - 573*x^3 - 222*x^2 + 123*x + 601)
 
gp: K = bnfinit(y^12 - y^11 - 4*y^10 - 10*y^9 + 39*y^8 - 78*y^7 + 214*y^6 - 280*y^5 + 693*y^4 - 573*y^3 - 222*y^2 + 123*y + 601, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 4*x^10 - 10*x^9 + 39*x^8 - 78*x^7 + 214*x^6 - 280*x^5 + 693*x^4 - 573*x^3 - 222*x^2 + 123*x + 601);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - x^11 - 4*x^10 - 10*x^9 + 39*x^8 - 78*x^7 + 214*x^6 - 280*x^5 + 693*x^4 - 573*x^3 - 222*x^2 + 123*x + 601)
 

\( x^{12} - x^{11} - 4 x^{10} - 10 x^{9} + 39 x^{8} - 78 x^{7} + 214 x^{6} - 280 x^{5} + 693 x^{4} - 573 x^{3} - 222 x^{2} + 123 x + 601 \) 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:   \(61132828589969773\) \(\medspace = 7^{8}\cdot 13^{9}\) 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:  \(25.05\)
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:  $7^{2/3}13^{3/4}\approx 25.052796318948193$
Ramified primes:   \(7\), \(13\) 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:  \(\Q(\sqrt{13}) \)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(91=7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{91}(64,·)$, $\chi_{91}(1,·)$, $\chi_{91}(8,·)$, $\chi_{91}(44,·)$, $\chi_{91}(79,·)$, $\chi_{91}(18,·)$, $\chi_{91}(51,·)$, $\chi_{91}(53,·)$, $\chi_{91}(86,·)$, $\chi_{91}(25,·)$, $\chi_{91}(57,·)$, $\chi_{91}(60,·)$$\rbrace$
This is a CM field.
Reflex fields:  4.0.2197.1$^{2}$, 12.0.61132828589969773.1$^{30}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{57\!\cdots\!97}a^{11}-\frac{11\!\cdots\!61}{19\!\cdots\!99}a^{10}-\frac{16\!\cdots\!53}{19\!\cdots\!99}a^{9}-\frac{74\!\cdots\!94}{19\!\cdots\!99}a^{8}+\frac{63\!\cdots\!70}{19\!\cdots\!99}a^{7}-\frac{86\!\cdots\!97}{19\!\cdots\!99}a^{6}+\frac{10\!\cdots\!12}{57\!\cdots\!97}a^{5}-\frac{38\!\cdots\!46}{19\!\cdots\!99}a^{4}-\frac{51\!\cdots\!15}{57\!\cdots\!97}a^{3}-\frac{15\!\cdots\!32}{57\!\cdots\!97}a^{2}+\frac{46\!\cdots\!21}{19\!\cdots\!99}a-\frac{53\!\cdots\!00}{19\!\cdots\!99}$ 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

$C_{2}\times C_{2}$, which has order $4$

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:   \( -1 \)  (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:   $\frac{13505158842215}{19\!\cdots\!99}a^{11}-\frac{10034830535733}{19\!\cdots\!99}a^{10}-\frac{68577174796381}{19\!\cdots\!99}a^{9}-\frac{182849101111416}{19\!\cdots\!99}a^{8}+\frac{444597967255710}{19\!\cdots\!99}a^{7}-\frac{646265040804596}{19\!\cdots\!99}a^{6}+\frac{33\!\cdots\!45}{19\!\cdots\!99}a^{5}-\frac{20\!\cdots\!14}{19\!\cdots\!99}a^{4}+\frac{53\!\cdots\!56}{19\!\cdots\!99}a^{3}-\frac{55\!\cdots\!87}{19\!\cdots\!99}a^{2}-\frac{99\!\cdots\!57}{19\!\cdots\!99}a-\frac{17\!\cdots\!22}{19\!\cdots\!99}$, $\frac{18606448286018}{19\!\cdots\!99}a^{11}-\frac{22787130098526}{19\!\cdots\!99}a^{10}-\frac{77885140992422}{19\!\cdots\!99}a^{9}-\frac{125483082135645}{19\!\cdots\!99}a^{8}+\frac{815682105662352}{19\!\cdots\!99}a^{7}-\frac{16\!\cdots\!38}{19\!\cdots\!99}a^{6}+\frac{34\!\cdots\!80}{19\!\cdots\!99}a^{5}-\frac{49\!\cdots\!53}{19\!\cdots\!99}a^{4}+\frac{11\!\cdots\!38}{19\!\cdots\!99}a^{3}-\frac{58\!\cdots\!17}{19\!\cdots\!99}a^{2}-\frac{14\!\cdots\!37}{19\!\cdots\!99}a+\frac{92\!\cdots\!88}{19\!\cdots\!99}$, $\frac{4759225}{36677252641}a^{11}-\frac{48283509}{36677252641}a^{10}+\frac{15475111}{110031757923}a^{9}+\frac{260541560}{36677252641}a^{8}+\frac{621682972}{36677252641}a^{7}-\frac{2377721180}{36677252641}a^{6}+\frac{2115942755}{36677252641}a^{5}-\frac{5373201551}{36677252641}a^{4}+\frac{15538474330}{110031757923}a^{3}+\frac{1914167001}{36677252641}a^{2}-\frac{3276057308}{110031757923}a+\frac{18037614890}{110031757923}$, $\frac{19779208541530}{57\!\cdots\!97}a^{11}-\frac{45518170004191}{57\!\cdots\!97}a^{10}-\frac{62087869413088}{19\!\cdots\!99}a^{9}-\frac{91412709544312}{19\!\cdots\!99}a^{8}+\frac{459363543766811}{19\!\cdots\!99}a^{7}-\frac{167454570157973}{19\!\cdots\!99}a^{6}+\frac{45\!\cdots\!20}{57\!\cdots\!97}a^{5}-\frac{36\!\cdots\!23}{57\!\cdots\!97}a^{4}+\frac{20\!\cdots\!72}{57\!\cdots\!97}a^{3}-\frac{17\!\cdots\!26}{57\!\cdots\!97}a^{2}-\frac{35\!\cdots\!22}{57\!\cdots\!97}a+\frac{10\!\cdots\!28}{19\!\cdots\!99}$, $\frac{4385715924758}{19\!\cdots\!99}a^{11}-\frac{9944953612049}{57\!\cdots\!97}a^{10}-\frac{39176104594902}{19\!\cdots\!99}a^{9}-\frac{79292545692662}{19\!\cdots\!99}a^{8}+\frac{245119790197250}{19\!\cdots\!99}a^{7}-\frac{62553440982499}{19\!\cdots\!99}a^{6}+\frac{883354903803517}{19\!\cdots\!99}a^{5}-\frac{20\!\cdots\!98}{57\!\cdots\!97}a^{4}+\frac{40\!\cdots\!58}{19\!\cdots\!99}a^{3}-\frac{10\!\cdots\!64}{57\!\cdots\!97}a^{2}-\frac{21\!\cdots\!23}{57\!\cdots\!97}a-\frac{78\!\cdots\!49}{19\!\cdots\!99}$ 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:  \( 562.775330001 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 562.775330001 \cdot 4}{2\cdot\sqrt{61132828589969773}}\cr\approx \mathstrut & 0.280096067652 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 4*x^10 - 10*x^9 + 39*x^8 - 78*x^7 + 214*x^6 - 280*x^5 + 693*x^4 - 573*x^3 - 222*x^2 + 123*x + 601)
 
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 - 4*x^10 - 10*x^9 + 39*x^8 - 78*x^7 + 214*x^6 - 280*x^5 + 693*x^4 - 573*x^3 - 222*x^2 + 123*x + 601, 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 - 4*x^10 - 10*x^9 + 39*x^8 - 78*x^7 + 214*x^6 - 280*x^5 + 693*x^4 - 573*x^3 - 222*x^2 + 123*x + 601);
 
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 - 4*x^10 - 10*x^9 + 39*x^8 - 78*x^7 + 214*x^6 - 280*x^5 + 693*x^4 - 573*x^3 - 222*x^2 + 123*x + 601);
 
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

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{7})^+\), 4.0.2197.1, 6.6.5274997.1

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 ${\href{/padicField/2.12.0.1}{12} }$ ${\href{/padicField/3.3.0.1}{3} }^{4}$ ${\href{/padicField/5.12.0.1}{12} }$ R ${\href{/padicField/11.12.0.1}{12} }$ R ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.12.0.1}{12} }$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.1.0.1}{1} }^{12}$ ${\href{/padicField/31.12.0.1}{12} }$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.4.0.1}{4} }^{3}$ ${\href{/padicField/43.2.0.1}{2} }^{6}$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.3.0.1}{3} }^{4}$ ${\href{/padicField/59.12.0.1}{12} }$

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}$
\(13\) Copy content Toggle raw display 13.4.3.2$x^{4} + 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} + 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} + 13$$4$$1$$3$$C_4$$[\ ]_{4}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.13.4t1.a.a$1$ $ 13 $ 4.0.2197.1 $C_4$ (as 4T1) $0$ $-1$
* 1.13.2t1.a.a$1$ $ 13 $ \(\Q(\sqrt{13}) \) $C_2$ (as 2T1) $1$ $1$
* 1.13.4t1.a.b$1$ $ 13 $ 4.0.2197.1 $C_4$ (as 4T1) $0$ $-1$
* 1.7.3t1.a.a$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
* 1.91.12t1.a.a$1$ $ 7 \cdot 13 $ 12.0.61132828589969773.1 $C_{12}$ (as 12T1) $0$ $-1$
* 1.91.6t1.j.a$1$ $ 7 \cdot 13 $ 6.6.5274997.1 $C_6$ (as 6T1) $0$ $1$
* 1.91.12t1.a.b$1$ $ 7 \cdot 13 $ 12.0.61132828589969773.1 $C_{12}$ (as 12T1) $0$ $-1$
* 1.7.3t1.a.b$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
* 1.91.12t1.a.c$1$ $ 7 \cdot 13 $ 12.0.61132828589969773.1 $C_{12}$ (as 12T1) $0$ $-1$
* 1.91.6t1.j.b$1$ $ 7 \cdot 13 $ 6.6.5274997.1 $C_6$ (as 6T1) $0$ $1$
* 1.91.12t1.a.d$1$ $ 7 \cdot 13 $ 12.0.61132828589969773.1 $C_{12}$ (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 *.