↑

Properties

Label 12.0.210...261.1
Degree $12$
Signature $[0, 6]$
Discriminant $2.110\times 10^{28}$
Root discriminant \(229.27\)
Ramified primes $17,37$
Class number $8788$ (GRH)
Class group [13, 13, 52] (GRH)
Galois group $C_{12}$ (as 12T1)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 76*x^10 + 316*x^9 + 2281*x^8 - 4273*x^7 + 296307*x^6 - 173877*x^5 + 8778862*x^4 + 7971726*x^3 + 102217297*x^2 + 117963773*x + 493326523)
 
gp: K = bnfinit(y^12 - y^11 + 76*y^10 + 316*y^9 + 2281*y^8 - 4273*y^7 + 296307*y^6 - 173877*y^5 + 8778862*y^4 + 7971726*y^3 + 102217297*y^2 + 117963773*y + 493326523, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 + 76*x^10 + 316*x^9 + 2281*x^8 - 4273*x^7 + 296307*x^6 - 173877*x^5 + 8778862*x^4 + 7971726*x^3 + 102217297*x^2 + 117963773*x + 493326523);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 + 76*x^10 + 316*x^9 + 2281*x^8 - 4273*x^7 + 296307*x^6 - 173877*x^5 + 8778862*x^4 + 7971726*x^3 + 102217297*x^2 + 117963773*x + 493326523)
 

\( x^{12} - x^{11} + 76 x^{10} + 316 x^{9} + 2281 x^{8} - 4273 x^{7} + 296307 x^{6} - 173877 x^{5} + \cdots + 493326523 \) 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:   \(21098872958222608828144613261\) \(\medspace = 17^{9}\cdot 37^{11}\) 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:  \(229.27\)
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:  $17^{3/4}37^{11/12}\approx 229.2740687298747$
Ramified primes:   \(17\), \(37\) 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:  \(\Q(\sqrt{629}) \)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(629=17\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{629}(1,·)$, $\chi_{629}(101,·)$, $\chi_{629}(137,·)$, $\chi_{629}(492,·)$, $\chi_{629}(191,·)$, $\chi_{629}(528,·)$, $\chi_{629}(628,·)$, $\chi_{629}(438,·)$, $\chi_{629}(378,·)$, $\chi_{629}(251,·)$, $\chi_{629}(208,·)$, $\chi_{629}(421,·)$$\rbrace$
This is a CM field.
Reflex fields:  4.0.248858189.2$^{2}$, 12.0.21098872958222608828144613261.1$^{30}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{47}a^{10}-\frac{12}{47}a^{9}-\frac{21}{47}a^{8}+\frac{5}{47}a^{7}-\frac{15}{47}a^{6}+\frac{11}{47}a^{5}-\frac{4}{47}a^{4}-\frac{8}{47}a^{3}-\frac{16}{47}a^{2}-\frac{4}{47}a$, $\frac{1}{56\!\cdots\!31}a^{11}+\frac{37\!\cdots\!32}{56\!\cdots\!31}a^{10}+\frac{25\!\cdots\!91}{12\!\cdots\!73}a^{9}-\frac{48\!\cdots\!90}{56\!\cdots\!31}a^{8}+\frac{17\!\cdots\!33}{56\!\cdots\!31}a^{7}-\frac{20\!\cdots\!34}{56\!\cdots\!31}a^{6}+\frac{19\!\cdots\!51}{56\!\cdots\!31}a^{5}-\frac{24\!\cdots\!26}{56\!\cdots\!31}a^{4}+\frac{93\!\cdots\!45}{56\!\cdots\!31}a^{3}-\frac{93\!\cdots\!62}{56\!\cdots\!31}a^{2}-\frac{77\!\cdots\!17}{56\!\cdots\!31}a-\frac{27\!\cdots\!00}{12\!\cdots\!73}$ 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

$C_{13}\times C_{13}\times C_{52}$, which has order $8788$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Relative class number: $4394$ (assuming GRH)

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:   $\frac{11\!\cdots\!48}{14\!\cdots\!63}a^{11}-\frac{50\!\cdots\!53}{14\!\cdots\!63}a^{10}+\frac{74\!\cdots\!57}{14\!\cdots\!63}a^{9}+\frac{16\!\cdots\!19}{14\!\cdots\!63}a^{8}+\frac{42\!\cdots\!17}{14\!\cdots\!63}a^{7}-\frac{12\!\cdots\!77}{14\!\cdots\!63}a^{6}+\frac{36\!\cdots\!39}{14\!\cdots\!63}a^{5}-\frac{87\!\cdots\!94}{14\!\cdots\!63}a^{4}+\frac{60\!\cdots\!76}{14\!\cdots\!63}a^{3}-\frac{63\!\cdots\!37}{14\!\cdots\!63}a^{2}+\frac{10\!\cdots\!79}{14\!\cdots\!63}a-\frac{20\!\cdots\!80}{31\!\cdots\!29}$, $\frac{11\!\cdots\!48}{14\!\cdots\!63}a^{11}-\frac{50\!\cdots\!53}{14\!\cdots\!63}a^{10}+\frac{74\!\cdots\!57}{14\!\cdots\!63}a^{9}+\frac{16\!\cdots\!19}{14\!\cdots\!63}a^{8}+\frac{42\!\cdots\!17}{14\!\cdots\!63}a^{7}-\frac{12\!\cdots\!77}{14\!\cdots\!63}a^{6}+\frac{36\!\cdots\!39}{14\!\cdots\!63}a^{5}-\frac{87\!\cdots\!94}{14\!\cdots\!63}a^{4}+\frac{60\!\cdots\!76}{14\!\cdots\!63}a^{3}-\frac{63\!\cdots\!37}{14\!\cdots\!63}a^{2}+\frac{10\!\cdots\!79}{14\!\cdots\!63}a+\frac{10\!\cdots\!49}{31\!\cdots\!29}$, $\frac{39\!\cdots\!43}{31\!\cdots\!27}a^{11}-\frac{56\!\cdots\!86}{31\!\cdots\!27}a^{10}+\frac{27\!\cdots\!61}{31\!\cdots\!27}a^{9}+\frac{11\!\cdots\!44}{31\!\cdots\!27}a^{8}+\frac{66\!\cdots\!16}{31\!\cdots\!27}a^{7}-\frac{23\!\cdots\!03}{31\!\cdots\!27}a^{6}+\frac{11\!\cdots\!92}{31\!\cdots\!27}a^{5}-\frac{10\!\cdots\!96}{31\!\cdots\!27}a^{4}+\frac{27\!\cdots\!02}{31\!\cdots\!27}a^{3}+\frac{41\!\cdots\!30}{31\!\cdots\!27}a^{2}+\frac{16\!\cdots\!13}{31\!\cdots\!27}a-\frac{13\!\cdots\!05}{31\!\cdots\!27}$, $\frac{60\!\cdots\!83}{62\!\cdots\!69}a^{11}-\frac{26\!\cdots\!89}{62\!\cdots\!69}a^{10}+\frac{39\!\cdots\!20}{62\!\cdots\!69}a^{9}+\frac{92\!\cdots\!27}{62\!\cdots\!69}a^{8}-\frac{16\!\cdots\!27}{62\!\cdots\!69}a^{7}-\frac{39\!\cdots\!20}{62\!\cdots\!69}a^{6}+\frac{31\!\cdots\!91}{13\!\cdots\!27}a^{5}-\frac{39\!\cdots\!42}{62\!\cdots\!69}a^{4}+\frac{24\!\cdots\!26}{62\!\cdots\!69}a^{3}-\frac{36\!\cdots\!31}{62\!\cdots\!69}a^{2}+\frac{18\!\cdots\!86}{62\!\cdots\!69}a-\frac{17\!\cdots\!41}{13\!\cdots\!27}$, $\frac{89\!\cdots\!26}{62\!\cdots\!69}a^{11}-\frac{73\!\cdots\!58}{62\!\cdots\!69}a^{10}+\frac{46\!\cdots\!96}{62\!\cdots\!69}a^{9}-\frac{11\!\cdots\!16}{62\!\cdots\!69}a^{8}-\frac{15\!\cdots\!01}{62\!\cdots\!69}a^{7}-\frac{22\!\cdots\!62}{62\!\cdots\!69}a^{6}+\frac{28\!\cdots\!20}{62\!\cdots\!69}a^{5}-\frac{16\!\cdots\!68}{62\!\cdots\!69}a^{4}+\frac{11\!\cdots\!52}{62\!\cdots\!69}a^{3}-\frac{31\!\cdots\!59}{62\!\cdots\!69}a^{2}-\frac{47\!\cdots\!38}{62\!\cdots\!69}a-\frac{45\!\cdots\!42}{13\!\cdots\!27}$ 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:  \( 92664.46751452335 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 92664.46751452335 \cdot 8788}{2\cdot\sqrt{21098872958222608828144613261}}\cr\approx \mathstrut & 0.172473695407446 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 76*x^10 + 316*x^9 + 2281*x^8 - 4273*x^7 + 296307*x^6 - 173877*x^5 + 8778862*x^4 + 7971726*x^3 + 102217297*x^2 + 117963773*x + 493326523)
 
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 + 76*x^10 + 316*x^9 + 2281*x^8 - 4273*x^7 + 296307*x^6 - 173877*x^5 + 8778862*x^4 + 7971726*x^3 + 102217297*x^2 + 117963773*x + 493326523, 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 + 76*x^10 + 316*x^9 + 2281*x^8 - 4273*x^7 + 296307*x^6 - 173877*x^5 + 8778862*x^4 + 7971726*x^3 + 102217297*x^2 + 117963773*x + 493326523);
 
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 + 76*x^10 + 316*x^9 + 2281*x^8 - 4273*x^7 + 296307*x^6 - 173877*x^5 + 8778862*x^4 + 7971726*x^3 + 102217297*x^2 + 117963773*x + 493326523);
 
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

\(\Q(\sqrt{629}) \), 3.3.1369.1, 4.0.248858189.2, 6.6.340686860741.1

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 ${\href{/padicField/2.12.0.1}{12} }$ ${\href{/padicField/3.12.0.1}{12} }$ ${\href{/padicField/5.3.0.1}{3} }^{4}$ ${\href{/padicField/7.12.0.1}{12} }$ ${\href{/padicField/11.4.0.1}{4} }^{3}$ ${\href{/padicField/13.12.0.1}{12} }$ R ${\href{/padicField/19.12.0.1}{12} }$ ${\href{/padicField/23.2.0.1}{2} }^{6}$ ${\href{/padicField/29.2.0.1}{2} }^{6}$ ${\href{/padicField/31.2.0.1}{2} }^{6}$ R ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.1.0.1}{1} }^{12}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.12.0.1}{12} }$

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
\(17\) Copy content Toggle raw display 17.12.9.3$x^{12} + 289 x^{4} - 68782$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
\(37\) Copy content Toggle raw display 37.12.11.11$x^{12} + 518$$12$$1$$11$$C_{12}$$[\ ]_{12}$