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Properties

Label 12.0.215...000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2.152\times 10^{21}$
Root discriminant \(59.94\)
Ramified primes $2,3,5,7$
Class number $1924$ (GRH)
Class group [2, 962] (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 - 4*x^11 + 88*x^10 - 276*x^9 + 3259*x^8 - 7972*x^7 + 64062*x^6 - 116656*x^5 + 705880*x^4 - 876724*x^3 + 4423460*x^2 - 2965388*x + 12267361)
 
gp: K = bnfinit(y^12 - 4*y^11 + 88*y^10 - 276*y^9 + 3259*y^8 - 7972*y^7 + 64062*y^6 - 116656*y^5 + 705880*y^4 - 876724*y^3 + 4423460*y^2 - 2965388*y + 12267361, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 88*x^10 - 276*x^9 + 3259*x^8 - 7972*x^7 + 64062*x^6 - 116656*x^5 + 705880*x^4 - 876724*x^3 + 4423460*x^2 - 2965388*x + 12267361);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 + 88*x^10 - 276*x^9 + 3259*x^8 - 7972*x^7 + 64062*x^6 - 116656*x^5 + 705880*x^4 - 876724*x^3 + 4423460*x^2 - 2965388*x + 12267361)
 

\( x^{12} - 4 x^{11} + 88 x^{10} - 276 x^{9} + 3259 x^{8} - 7972 x^{7} + 64062 x^{6} - 116656 x^{5} + \cdots + 12267361 \) 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:   \(2151700443648000000000\) \(\medspace = 2^{18}\cdot 3^{6}\cdot 5^{9}\cdot 7^{8}\) 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:  \(59.94\)
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:  $2^{3/2}3^{1/2}5^{3/4}7^{2/3}\approx 59.94208117178477$
Ramified primes:   \(2\), \(3\), \(5\), \(7\) 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:  \(840=2^{3}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(323,·)$, $\chi_{840}(289,·)$, $\chi_{840}(169,·)$, $\chi_{840}(683,·)$, $\chi_{840}(827,·)$, $\chi_{840}(529,·)$, $\chi_{840}(107,·)$, $\chi_{840}(361,·)$, $\chi_{840}(121,·)$, $\chi_{840}(347,·)$, $\chi_{840}(443,·)$$\rbrace$
This is a CM field.
Reflex fields:  4.0.72000.2$^{2}$, 12.0.2151700443648000000000.1$^{30}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6}a^{6}-\frac{1}{3}a^{5}-\frac{1}{2}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{6}$, $\frac{1}{6}a^{7}-\frac{1}{6}a^{5}+\frac{1}{3}a^{3}-\frac{1}{6}a+\frac{1}{3}$, $\frac{1}{6}a^{8}-\frac{1}{3}a^{5}-\frac{1}{6}a^{4}+\frac{1}{6}a^{2}-\frac{1}{3}a+\frac{1}{6}$, $\frac{1}{6}a^{9}+\frac{1}{6}a^{5}+\frac{1}{6}a^{3}+\frac{1}{3}a^{2}-\frac{1}{6}a+\frac{1}{3}$, $\frac{1}{523090086}a^{10}-\frac{7784673}{174363362}a^{9}+\frac{4010603}{87181681}a^{8}-\frac{6887283}{174363362}a^{7}-\frac{25129711}{523090086}a^{6}+\frac{19144715}{261545043}a^{5}+\frac{112177363}{523090086}a^{4}-\frac{54857551}{523090086}a^{3}-\frac{37048355}{523090086}a^{2}+\frac{33284309}{261545043}a-\frac{115934053}{261545043}$, $\frac{1}{21\!\cdots\!26}a^{11}+\frac{8513813668}{10\!\cdots\!63}a^{10}-\frac{23\!\cdots\!37}{21\!\cdots\!26}a^{9}-\frac{36\!\cdots\!78}{10\!\cdots\!63}a^{8}+\frac{86\!\cdots\!99}{10\!\cdots\!63}a^{7}+\frac{53\!\cdots\!21}{71\!\cdots\!42}a^{6}-\frac{99\!\cdots\!99}{21\!\cdots\!26}a^{5}-\frac{45\!\cdots\!07}{21\!\cdots\!26}a^{4}-\frac{46\!\cdots\!25}{35\!\cdots\!21}a^{3}-\frac{10\!\cdots\!32}{35\!\cdots\!21}a^{2}+\frac{91\!\cdots\!12}{35\!\cdots\!21}a+\frac{55\!\cdots\!23}{21\!\cdots\!26}$ 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

$C_{2}\times C_{962}$, which has order $1924$ (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:  $5$
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:   $\frac{14}{261545043}a^{11}+\frac{925}{261545043}a^{10}-\frac{705}{87181681}a^{9}+\frac{126965}{523090086}a^{8}-\frac{43800}{87181681}a^{7}+\frac{1759051}{261545043}a^{6}-\frac{2862538}{261545043}a^{5}+\frac{47323775}{523090086}a^{4}-\frac{26117620}{261545043}a^{3}+\frac{349531301}{523090086}a^{2}-\frac{105854134}{261545043}a+\frac{660072195}{174363362}$, $\frac{202473537260}{35\!\cdots\!21}a^{11}+\frac{13037056344926}{35\!\cdots\!21}a^{10}-\frac{37427947472440}{35\!\cdots\!21}a^{9}+\frac{11\!\cdots\!50}{35\!\cdots\!21}a^{8}-\frac{30\!\cdots\!20}{35\!\cdots\!21}a^{7}+\frac{40\!\cdots\!30}{35\!\cdots\!21}a^{6}-\frac{82\!\cdots\!20}{35\!\cdots\!21}a^{5}+\frac{71\!\cdots\!05}{35\!\cdots\!21}a^{4}-\frac{10\!\cdots\!40}{35\!\cdots\!21}a^{3}+\frac{62\!\cdots\!60}{35\!\cdots\!21}a^{2}-\frac{47\!\cdots\!20}{35\!\cdots\!21}a+\frac{24\!\cdots\!08}{35\!\cdots\!21}$, $\frac{202473537260}{35\!\cdots\!21}a^{11}+\frac{13037056344926}{35\!\cdots\!21}a^{10}-\frac{37427947472440}{35\!\cdots\!21}a^{9}+\frac{11\!\cdots\!50}{35\!\cdots\!21}a^{8}-\frac{30\!\cdots\!20}{35\!\cdots\!21}a^{7}+\frac{40\!\cdots\!30}{35\!\cdots\!21}a^{6}-\frac{82\!\cdots\!20}{35\!\cdots\!21}a^{5}+\frac{71\!\cdots\!05}{35\!\cdots\!21}a^{4}-\frac{10\!\cdots\!40}{35\!\cdots\!21}a^{3}+\frac{62\!\cdots\!60}{35\!\cdots\!21}a^{2}-\frac{47\!\cdots\!20}{35\!\cdots\!21}a+\frac{28\!\cdots\!29}{35\!\cdots\!21}$, $\frac{5028155171174}{10\!\cdots\!63}a^{11}+\frac{20264568865213}{10\!\cdots\!63}a^{10}+\frac{54469677253855}{35\!\cdots\!21}a^{9}+\frac{36\!\cdots\!55}{21\!\cdots\!26}a^{8}-\frac{268201050564200}{35\!\cdots\!21}a^{7}+\frac{59\!\cdots\!71}{10\!\cdots\!63}a^{6}-\frac{11\!\cdots\!86}{10\!\cdots\!63}a^{5}+\frac{18\!\cdots\!55}{21\!\cdots\!26}a^{4}-\frac{19\!\cdots\!20}{10\!\cdots\!63}a^{3}+\frac{15\!\cdots\!81}{21\!\cdots\!26}a^{2}-\frac{98\!\cdots\!54}{10\!\cdots\!63}a+\frac{22\!\cdots\!41}{71\!\cdots\!42}$, $\frac{34099247206}{10\!\cdots\!63}a^{11}+\frac{1231007446853}{10\!\cdots\!63}a^{10}-\frac{8557121613535}{35\!\cdots\!21}a^{9}+\frac{17\!\cdots\!35}{21\!\cdots\!26}a^{8}-\frac{12\!\cdots\!20}{35\!\cdots\!21}a^{7}+\frac{50\!\cdots\!99}{10\!\cdots\!63}a^{6}-\frac{13\!\cdots\!02}{10\!\cdots\!63}a^{5}+\frac{23\!\cdots\!55}{21\!\cdots\!26}a^{4}-\frac{19\!\cdots\!00}{10\!\cdots\!63}a^{3}+\frac{22\!\cdots\!19}{21\!\cdots\!26}a^{2}-\frac{98\!\cdots\!66}{10\!\cdots\!63}a+\frac{29\!\cdots\!63}{71\!\cdots\!42}$ 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:  \( 104.882003477 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 104.882003477 \cdot 1924}{2\cdot\sqrt{2151700443648000000000}}\cr\approx \mathstrut & 0.133833351000 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 88*x^10 - 276*x^9 + 3259*x^8 - 7972*x^7 + 64062*x^6 - 116656*x^5 + 705880*x^4 - 876724*x^3 + 4423460*x^2 - 2965388*x + 12267361)
 
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 + 88*x^10 - 276*x^9 + 3259*x^8 - 7972*x^7 + 64062*x^6 - 116656*x^5 + 705880*x^4 - 876724*x^3 + 4423460*x^2 - 2965388*x + 12267361, 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 + 88*x^10 - 276*x^9 + 3259*x^8 - 7972*x^7 + 64062*x^6 - 116656*x^5 + 705880*x^4 - 876724*x^3 + 4423460*x^2 - 2965388*x + 12267361);
 
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 + 88*x^10 - 276*x^9 + 3259*x^8 - 7972*x^7 + 64062*x^6 - 116656*x^5 + 705880*x^4 - 876724*x^3 + 4423460*x^2 - 2965388*x + 12267361);
 
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.72000.2, 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 R R R ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{3}$ ${\href{/padicField/17.12.0.1}{12} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.2.0.1}{2} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.12.0.1}{12} }$ ${\href{/padicField/59.3.0.1}{3} }^{4}$

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 2.12.18.27$x^{12} + 24 x^{11} + 300 x^{10} + 2480 x^{9} + 15084 x^{8} + 70848 x^{7} + 263968 x^{6} + 785280 x^{5} + 1858672 x^{4} + 3423104 x^{3} + 4742336 x^{2} + 4511488 x + 2639680$$2$$6$$18$$C_{12}$$[3]^{6}$
\(3\) Copy content Toggle raw display 3.12.6.1$x^{12} + 18 x^{8} + 81 x^{4} - 486 x^{2} + 1458$$2$$6$$6$$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}$