↑

Properties

Label 12.0.160...944.2
Degree $12$
Signature $[0, 6]$
Discriminant $1.601\times 10^{25}$
Root discriminant \(126.00\)
Ramified primes $2,3,53$
Class number $127008$ (GRH)
Class group [2, 252, 252] (GRH)
Galois group $C_6\times C_2$ (as 12T2)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 144*x^10 - 4*x^9 + 9417*x^8 + 36*x^7 + 354874*x^6 + 17724*x^5 + 8053932*x^4 + 638028*x^3 + 103405548*x^2 + 7102716*x + 582677873)
 
gp: K = bnfinit(y^12 + 144*y^10 - 4*y^9 + 9417*y^8 + 36*y^7 + 354874*y^6 + 17724*y^5 + 8053932*y^4 + 638028*y^3 + 103405548*y^2 + 7102716*y + 582677873, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 144*x^10 - 4*x^9 + 9417*x^8 + 36*x^7 + 354874*x^6 + 17724*x^5 + 8053932*x^4 + 638028*x^3 + 103405548*x^2 + 7102716*x + 582677873);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 + 144*x^10 - 4*x^9 + 9417*x^8 + 36*x^7 + 354874*x^6 + 17724*x^5 + 8053932*x^4 + 638028*x^3 + 103405548*x^2 + 7102716*x + 582677873)
 

\( x^{12} + 144 x^{10} - 4 x^{9} + 9417 x^{8} + 36 x^{7} + 354874 x^{6} + 17724 x^{5} + 8053932 x^{4} + \cdots + 582677873 \) 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:   \(16007193286004365741522944\) \(\medspace = 2^{24}\cdot 3^{16}\cdot 53^{6}\) 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:  \(126.00\)
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:  $2^{2}3^{4/3}53^{1/2}\approx 125.99682431526651$
Ramified primes:   \(2\), \(3\), \(53\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3816=2^{3}\cdot 3^{2}\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{3816}(1,·)$, $\chi_{3816}(2755,·)$, $\chi_{3816}(2119,·)$, $\chi_{3816}(1483,·)$, $\chi_{3816}(3181,·)$, $\chi_{3816}(847,·)$, $\chi_{3816}(2545,·)$, $\chi_{3816}(211,·)$, $\chi_{3816}(1909,·)$, $\chi_{3816}(1273,·)$, $\chi_{3816}(637,·)$, $\chi_{3816}(3391,·)$$\rbrace$
This is a CM field.
Reflex fields:  \(\Q(\sqrt{-53}) \), \(\Q(\sqrt{-106}) \), 6.0.62514047808.3$^{3}$, 6.0.500112382464.5$^{3}$, 12.0.16007193286004365741522944.2$^{24}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{81}a^{9}-\frac{5}{27}a^{7}-\frac{1}{27}a^{6}-\frac{8}{27}a^{5}+\frac{4}{27}a^{4}+\frac{19}{81}a^{3}-\frac{1}{27}a^{2}+\frac{1}{9}a+\frac{1}{81}$, $\frac{1}{1930458987}a^{10}-\frac{2649521}{1930458987}a^{9}-\frac{50469458}{643486329}a^{8}-\frac{3303565}{23832827}a^{7}+\frac{49952503}{214495443}a^{6}+\frac{62057810}{643486329}a^{5}-\frac{39170992}{113556411}a^{4}+\frac{181166413}{1930458987}a^{3}+\frac{157945154}{643486329}a^{2}-\frac{348608627}{1930458987}a-\frac{55859659}{113556411}$, $\frac{1}{15\!\cdots\!09}a^{11}+\frac{22379608387}{15\!\cdots\!09}a^{10}-\frac{22\!\cdots\!57}{90\!\cdots\!77}a^{9}-\frac{15\!\cdots\!69}{17\!\cdots\!01}a^{8}-\frac{11\!\cdots\!37}{51\!\cdots\!03}a^{7}+\frac{20\!\cdots\!70}{51\!\cdots\!03}a^{6}+\frac{34\!\cdots\!14}{15\!\cdots\!09}a^{5}+\frac{30\!\cdots\!51}{15\!\cdots\!09}a^{4}-\frac{50\!\cdots\!66}{15\!\cdots\!09}a^{3}+\frac{32\!\cdots\!50}{15\!\cdots\!09}a^{2}+\frac{70\!\cdots\!98}{15\!\cdots\!09}a-\frac{43\!\cdots\!99}{90\!\cdots\!77}$ 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

$C_{2}\times C_{252}\times C_{252}$, which has order $127008$ (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:   $\frac{8082873989188}{57\!\cdots\!67}a^{11}+\frac{9719173288066}{57\!\cdots\!67}a^{10}+\frac{913292221655284}{57\!\cdots\!67}a^{9}+\frac{152612053138017}{63\!\cdots\!63}a^{8}+\frac{15\!\cdots\!96}{19\!\cdots\!89}a^{7}+\frac{28\!\cdots\!54}{19\!\cdots\!89}a^{6}+\frac{13\!\cdots\!76}{57\!\cdots\!67}a^{5}+\frac{28\!\cdots\!85}{57\!\cdots\!67}a^{4}+\frac{20\!\cdots\!28}{57\!\cdots\!67}a^{3}+\frac{50\!\cdots\!52}{57\!\cdots\!67}a^{2}+\frac{13\!\cdots\!96}{57\!\cdots\!67}a+\frac{21\!\cdots\!68}{33\!\cdots\!51}$, $\frac{2744310247648}{57\!\cdots\!67}a^{11}+\frac{8936203299940}{57\!\cdots\!67}a^{10}+\frac{311734267233712}{57\!\cdots\!67}a^{9}+\frac{142466127543863}{63\!\cdots\!63}a^{8}+\frac{53\!\cdots\!44}{19\!\cdots\!89}a^{7}+\frac{26\!\cdots\!12}{19\!\cdots\!89}a^{6}+\frac{46\!\cdots\!08}{57\!\cdots\!67}a^{5}+\frac{26\!\cdots\!46}{57\!\cdots\!67}a^{4}+\frac{74\!\cdots\!20}{57\!\cdots\!67}a^{3}+\frac{46\!\cdots\!88}{57\!\cdots\!67}a^{2}+\frac{52\!\cdots\!00}{57\!\cdots\!67}a+\frac{21\!\cdots\!13}{33\!\cdots\!51}$, $\frac{102958718}{110309942503539}a^{11}+\frac{344263637}{110309942503539}a^{10}+\frac{11602140791}{110309942503539}a^{9}+\frac{5489717416}{12256660278171}a^{8}+\frac{197201007818}{36769980834513}a^{7}+\frac{1008026512286}{36769980834513}a^{6}+\frac{16715216680235}{110309942503539}a^{5}+\frac{100655777121464}{110309942503539}a^{4}+\frac{257827018870451}{110309942503539}a^{3}+\frac{18\!\cdots\!91}{110309942503539}a^{2}+\frac{17\!\cdots\!81}{110309942503539}a+\frac{828368918623360}{6488820147267}$, $\frac{73896579223618}{15\!\cdots\!09}a^{11}-\frac{220216186105265}{15\!\cdots\!09}a^{10}+\frac{83\!\cdots\!47}{15\!\cdots\!09}a^{9}-\frac{35\!\cdots\!37}{17\!\cdots\!01}a^{8}+\frac{14\!\cdots\!34}{51\!\cdots\!03}a^{7}-\frac{64\!\cdots\!08}{51\!\cdots\!03}a^{6}+\frac{12\!\cdots\!67}{15\!\cdots\!09}a^{5}-\frac{63\!\cdots\!89}{15\!\cdots\!09}a^{4}+\frac{19\!\cdots\!75}{15\!\cdots\!09}a^{3}-\frac{11\!\cdots\!17}{15\!\cdots\!09}a^{2}+\frac{12\!\cdots\!81}{15\!\cdots\!09}a-\frac{55\!\cdots\!47}{90\!\cdots\!77}$, $\frac{147992955910114}{15\!\cdots\!09}a^{11}+\frac{21061302993115}{15\!\cdots\!09}a^{10}+\frac{16\!\cdots\!71}{15\!\cdots\!09}a^{9}+\frac{270905851680464}{17\!\cdots\!01}a^{8}+\frac{28\!\cdots\!22}{51\!\cdots\!03}a^{7}+\frac{65\!\cdots\!16}{51\!\cdots\!03}a^{6}+\frac{24\!\cdots\!83}{15\!\cdots\!09}a^{5}+\frac{69\!\cdots\!53}{15\!\cdots\!09}a^{4}+\frac{39\!\cdots\!15}{15\!\cdots\!09}a^{3}+\frac{10\!\cdots\!59}{15\!\cdots\!09}a^{2}+\frac{26\!\cdots\!81}{15\!\cdots\!09}a+\frac{17\!\cdots\!04}{90\!\cdots\!77}$ 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:  \( 481.70037561485367 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 481.70037561485367 \cdot 127008}{2\cdot\sqrt{16007193286004365741522944}}\cr\approx \mathstrut & 0.470435061423374 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 + 144*x^10 - 4*x^9 + 9417*x^8 + 36*x^7 + 354874*x^6 + 17724*x^5 + 8053932*x^4 + 638028*x^3 + 103405548*x^2 + 7102716*x + 582677873)
 
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 + 144*x^10 - 4*x^9 + 9417*x^8 + 36*x^7 + 354874*x^6 + 17724*x^5 + 8053932*x^4 + 638028*x^3 + 103405548*x^2 + 7102716*x + 582677873, 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 + 144*x^10 - 4*x^9 + 9417*x^8 + 36*x^7 + 354874*x^6 + 17724*x^5 + 8053932*x^4 + 638028*x^3 + 103405548*x^2 + 7102716*x + 582677873);
 
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 + 144*x^10 - 4*x^9 + 9417*x^8 + 36*x^7 + 354874*x^6 + 17724*x^5 + 8053932*x^4 + 638028*x^3 + 103405548*x^2 + 7102716*x + 582677873);
 
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

$C_2\times C_6$ (as 12T2):

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)
 
An abelian group of order 12
The 12 conjugacy class representatives for $C_6\times C_2$
Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{-106}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-53}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{2}, \sqrt{-53})\), 6.0.500112382464.5, 6.6.3359232.1, 6.0.62514047808.3

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 ${\href{/padicField/5.6.0.1}{6} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.1.0.1}{1} }^{12}$ ${\href{/padicField/19.2.0.1}{2} }^{6}$ ${\href{/padicField/23.3.0.1}{3} }^{4}$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{6}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ R ${\href{/padicField/59.6.0.1}{6} }^{2}$

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
\(2\) Copy content Toggle raw display 2.12.24.318$x^{12} + 10 x^{10} + 12 x^{9} + 110 x^{8} + 80 x^{7} + 752 x^{6} + 512 x^{5} + 1636 x^{4} + 1504 x^{3} + 1224 x^{2} + 1008 x - 648$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
\(3\) Copy content Toggle raw display 3.6.8.3$x^{6} + 18 x^{5} + 114 x^{4} + 326 x^{3} + 570 x^{2} + 528 x + 197$$3$$2$$8$$C_6$$[2]^{2}$
3.6.8.3$x^{6} + 18 x^{5} + 114 x^{4} + 326 x^{3} + 570 x^{2} + 528 x + 197$$3$$2$$8$$C_6$$[2]^{2}$
\(53\) Copy content Toggle raw display 53.4.2.1$x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$