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Properties

Label 12.0.970...125.1
Degree $12$
Signature $[0, 6]$
Discriminant $9.708\times 10^{24}$
Root discriminant \(120.85\)
Ramified primes $3,5,7,17$
Class number $143624$ (GRH)
Class group [2, 2, 35906] (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 - x^11 + 443*x^10 - 444*x^9 + 72854*x^8 - 73298*x^7 + 5685112*x^6 - 6113764*x^5 + 223464715*x^4 - 256778386*x^3 + 4246142315*x^2 - 4044668347*x + 31633408171)
 
gp: K = bnfinit(y^12 - y^11 + 443*y^10 - 444*y^9 + 72854*y^8 - 73298*y^7 + 5685112*y^6 - 6113764*y^5 + 223464715*y^4 - 256778386*y^3 + 4246142315*y^2 - 4044668347*y + 31633408171, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 + 443*x^10 - 444*x^9 + 72854*x^8 - 73298*x^7 + 5685112*x^6 - 6113764*x^5 + 223464715*x^4 - 256778386*x^3 + 4246142315*x^2 - 4044668347*x + 31633408171);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 + 443*x^10 - 444*x^9 + 72854*x^8 - 73298*x^7 + 5685112*x^6 - 6113764*x^5 + 223464715*x^4 - 256778386*x^3 + 4246142315*x^2 - 4044668347*x + 31633408171)
 

\( x^{12} - x^{11} + 443 x^{10} - 444 x^{9} + 72854 x^{8} - 73298 x^{7} + 5685112 x^{6} + \cdots + 31633408171 \) 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:   \(9708038629029565330078125\) \(\medspace = 3^{6}\cdot 5^{9}\cdot 7^{10}\cdot 17^{6}\) 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:  \(120.85\)
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:  $3^{1/2}5^{3/4}7^{5/6}17^{1/2}\approx 120.85398018991815$
Ramified primes:   \(3\), \(5\), \(7\), \(17\) 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:  \(1785=3\cdot 5\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{1785}(256,·)$, $\chi_{1785}(1,·)$, $\chi_{1785}(1223,·)$, $\chi_{1785}(713,·)$, $\chi_{1785}(458,·)$, $\chi_{1785}(1427,·)$, $\chi_{1785}(1172,·)$, $\chi_{1785}(1429,·)$, $\chi_{1785}(919,·)$, $\chi_{1785}(152,·)$, $\chi_{1785}(1684,·)$, $\chi_{1785}(1276,·)$$\rbrace$
This is a CM field.
Reflex fields:  4.0.15931125.1$^{2}$, 12.0.9708038629029565330078125.1$^{30}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{1024846549}a^{10}+\frac{114206734}{1024846549}a^{9}-\frac{171396636}{1024846549}a^{8}+\frac{108291908}{1024846549}a^{7}+\frac{41425790}{1024846549}a^{6}+\frac{102298999}{1024846549}a^{5}-\frac{475518299}{1024846549}a^{4}-\frac{117684046}{1024846549}a^{3}+\frac{96287745}{1024846549}a^{2}-\frac{42480026}{1024846549}a+\frac{387181469}{1024846549}$, $\frac{1}{58\!\cdots\!21}a^{11}+\frac{26\!\cdots\!45}{58\!\cdots\!21}a^{10}+\frac{19\!\cdots\!78}{58\!\cdots\!21}a^{9}+\frac{10\!\cdots\!85}{58\!\cdots\!21}a^{8}+\frac{24\!\cdots\!32}{58\!\cdots\!21}a^{7}-\frac{11\!\cdots\!20}{83\!\cdots\!51}a^{6}+\frac{25\!\cdots\!55}{83\!\cdots\!51}a^{5}-\frac{27\!\cdots\!39}{58\!\cdots\!21}a^{4}-\frac{25\!\cdots\!77}{58\!\cdots\!21}a^{3}+\frac{17\!\cdots\!63}{58\!\cdots\!21}a^{2}+\frac{25\!\cdots\!48}{58\!\cdots\!21}a-\frac{38\!\cdots\!08}{18\!\cdots\!51}$ 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

$C_{2}\times C_{2}\times C_{35906}$, which has order $143624$ (assuming GRH)

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

Relative class number: $143624$ (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:   $\frac{16\!\cdots\!19}{57\!\cdots\!29}a^{11}-\frac{77\!\cdots\!96}{57\!\cdots\!29}a^{10}+\frac{66\!\cdots\!31}{57\!\cdots\!29}a^{9}-\frac{31\!\cdots\!66}{57\!\cdots\!29}a^{8}+\frac{99\!\cdots\!50}{57\!\cdots\!29}a^{7}-\frac{63\!\cdots\!59}{81\!\cdots\!99}a^{6}+\frac{91\!\cdots\!37}{81\!\cdots\!99}a^{5}-\frac{27\!\cdots\!00}{57\!\cdots\!29}a^{4}+\frac{19\!\cdots\!93}{57\!\cdots\!29}a^{3}-\frac{71\!\cdots\!74}{57\!\cdots\!29}a^{2}+\frac{21\!\cdots\!78}{57\!\cdots\!29}a-\frac{16\!\cdots\!52}{18\!\cdots\!99}$, $\frac{56\!\cdots\!30}{13\!\cdots\!29}a^{11}-\frac{19\!\cdots\!66}{13\!\cdots\!29}a^{10}+\frac{28\!\cdots\!50}{13\!\cdots\!29}a^{9}-\frac{75\!\cdots\!05}{13\!\cdots\!29}a^{8}+\frac{52\!\cdots\!05}{13\!\cdots\!29}a^{7}-\frac{14\!\cdots\!20}{18\!\cdots\!99}a^{6}+\frac{59\!\cdots\!15}{18\!\cdots\!99}a^{5}-\frac{55\!\cdots\!70}{13\!\cdots\!29}a^{4}+\frac{15\!\cdots\!05}{13\!\cdots\!29}a^{3}-\frac{12\!\cdots\!90}{13\!\cdots\!29}a^{2}+\frac{21\!\cdots\!65}{13\!\cdots\!29}a-\frac{32\!\cdots\!48}{41\!\cdots\!99}$, $\frac{56\!\cdots\!30}{13\!\cdots\!29}a^{11}-\frac{19\!\cdots\!66}{13\!\cdots\!29}a^{10}+\frac{28\!\cdots\!50}{13\!\cdots\!29}a^{9}-\frac{75\!\cdots\!05}{13\!\cdots\!29}a^{8}+\frac{52\!\cdots\!05}{13\!\cdots\!29}a^{7}-\frac{14\!\cdots\!20}{18\!\cdots\!99}a^{6}+\frac{59\!\cdots\!15}{18\!\cdots\!99}a^{5}-\frac{55\!\cdots\!70}{13\!\cdots\!29}a^{4}+\frac{15\!\cdots\!05}{13\!\cdots\!29}a^{3}-\frac{12\!\cdots\!90}{13\!\cdots\!29}a^{2}+\frac{21\!\cdots\!65}{13\!\cdots\!29}a-\frac{28\!\cdots\!49}{41\!\cdots\!99}$, $\frac{13\!\cdots\!61}{58\!\cdots\!21}a^{11}+\frac{81\!\cdots\!30}{58\!\cdots\!21}a^{10}+\frac{55\!\cdots\!69}{58\!\cdots\!21}a^{9}+\frac{13\!\cdots\!11}{58\!\cdots\!21}a^{8}+\frac{78\!\cdots\!05}{58\!\cdots\!21}a^{7}-\frac{21\!\cdots\!11}{83\!\cdots\!51}a^{6}+\frac{66\!\cdots\!78}{83\!\cdots\!51}a^{5}-\frac{37\!\cdots\!70}{58\!\cdots\!21}a^{4}+\frac{12\!\cdots\!12}{58\!\cdots\!21}a^{3}-\frac{14\!\cdots\!16}{58\!\cdots\!21}a^{2}+\frac{12\!\cdots\!37}{58\!\cdots\!21}a-\frac{24\!\cdots\!96}{18\!\cdots\!51}$, $\frac{31\!\cdots\!96}{58\!\cdots\!21}a^{11}-\frac{21\!\cdots\!78}{58\!\cdots\!21}a^{10}+\frac{13\!\cdots\!19}{58\!\cdots\!21}a^{9}-\frac{86\!\cdots\!69}{58\!\cdots\!21}a^{8}+\frac{19\!\cdots\!95}{58\!\cdots\!21}a^{7}-\frac{16\!\cdots\!96}{83\!\cdots\!51}a^{6}+\frac{18\!\cdots\!59}{83\!\cdots\!51}a^{5}-\frac{71\!\cdots\!80}{58\!\cdots\!21}a^{4}+\frac{38\!\cdots\!27}{58\!\cdots\!21}a^{3}-\frac{17\!\cdots\!61}{58\!\cdots\!21}a^{2}+\frac{44\!\cdots\!07}{58\!\cdots\!21}a-\frac{35\!\cdots\!79}{18\!\cdots\!51}$ 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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 104.882003477 \cdot 143624}{2\cdot\sqrt{9708038629029565330078125}}\cr\approx \mathstrut & 0.148734411436 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 443*x^10 - 444*x^9 + 72854*x^8 - 73298*x^7 + 5685112*x^6 - 6113764*x^5 + 223464715*x^4 - 256778386*x^3 + 4246142315*x^2 - 4044668347*x + 31633408171)
 
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 + 443*x^10 - 444*x^9 + 72854*x^8 - 73298*x^7 + 5685112*x^6 - 6113764*x^5 + 223464715*x^4 - 256778386*x^3 + 4246142315*x^2 - 4044668347*x + 31633408171, 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 + 443*x^10 - 444*x^9 + 72854*x^8 - 73298*x^7 + 5685112*x^6 - 6113764*x^5 + 223464715*x^4 - 256778386*x^3 + 4246142315*x^2 - 4044668347*x + 31633408171);
 
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 + 443*x^10 - 444*x^9 + 72854*x^8 - 73298*x^7 + 5685112*x^6 - 6113764*x^5 + 223464715*x^4 - 256778386*x^3 + 4246142315*x^2 - 4044668347*x + 31633408171);
 
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.15931125.1, 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 ${\href{/padicField/2.12.0.1}{12} }$ R R R ${\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{3}$ R ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.2.0.1}{2} }^{6}$ ${\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.12.0.1}{12} }$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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
\(3\) 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 5.12.9.1$x^{12} - 30 x^{8} + 225 x^{4} + 1125$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
\(7\) Copy content Toggle raw display 7.12.10.5$x^{12} - 154 x^{6} - 1421$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
\(17\) Copy content Toggle raw display 17.12.6.2$x^{12} + 578 x^{8} + 835210 x^{4} - 4259571 x^{2} + 72412707$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$