↑

Properties

Label 12.0.246...625.2
Degree $12$
Signature $[0, 6]$
Discriminant $2.462\times 10^{23}$
Root discriminant \(88.98\)
Ramified primes $3,5,43$
Class number $16226$ (GRH)
Class group [16226] (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 - x^11 + 62*x^10 - 226*x^9 + 577*x^8 - 2377*x^7 - 3136*x^6 + 12956*x^5 + 79168*x^4 + 61904*x^3 + 194304*x^2 + 378688*x + 235264)
 
gp: K = bnfinit(y^12 - y^11 + 62*y^10 - 226*y^9 + 577*y^8 - 2377*y^7 - 3136*y^6 + 12956*y^5 + 79168*y^4 + 61904*y^3 + 194304*y^2 + 378688*y + 235264, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 + 62*x^10 - 226*x^9 + 577*x^8 - 2377*x^7 - 3136*x^6 + 12956*x^5 + 79168*x^4 + 61904*x^3 + 194304*x^2 + 378688*x + 235264);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - x^11 + 62*x^10 - 226*x^9 + 577*x^8 - 2377*x^7 - 3136*x^6 + 12956*x^5 + 79168*x^4 + 61904*x^3 + 194304*x^2 + 378688*x + 235264)
 

\( x^{12} - x^{11} + 62 x^{10} - 226 x^{9} + 577 x^{8} - 2377 x^{7} - 3136 x^{6} + 12956 x^{5} + \cdots + 235264 \) 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:   \(246168290724753398765625\) \(\medspace = 3^{6}\cdot 5^{6}\cdot 43^{10}\) 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:  \(88.98\)
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:  $3^{1/2}5^{1/2}43^{5/6}\approx 88.97527572500881$
Ramified primes:   \(3\), \(5\), \(43\) 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:  \(645=3\cdot 5\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{645}(1,·)$, $\chi_{645}(386,·)$, $\chi_{645}(214,·)$, $\chi_{645}(424,·)$, $\chi_{645}(394,·)$, $\chi_{645}(44,·)$, $\chi_{645}(466,·)$, $\chi_{645}(436,·)$, $\chi_{645}(566,·)$, $\chi_{645}(596,·)$, $\chi_{645}(509,·)$, $\chi_{645}(479,·)$$\rbrace$
This is a CM field.
Reflex fields:  \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-215}) \), 6.0.11538453375.2$^{3}$, 6.0.18376055375.1$^{3}$, 12.0.246168290724753398765625.2$^{24}$

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

$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{16}a^{5}-\frac{1}{16}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a$, $\frac{1}{32}a^{6}-\frac{1}{32}a^{5}-\frac{1}{32}a^{4}+\frac{5}{32}a^{3}+\frac{3}{8}a-\frac{1}{2}$, $\frac{1}{64}a^{7}-\frac{1}{32}a^{5}-\frac{1}{16}a^{4}+\frac{5}{64}a^{3}-\frac{3}{16}a^{2}-\frac{1}{16}a+\frac{1}{4}$, $\frac{1}{768}a^{8}-\frac{1}{384}a^{7}+\frac{5}{384}a^{6}-\frac{1}{64}a^{5}-\frac{31}{768}a^{4}-\frac{61}{384}a^{3}+\frac{7}{64}a^{2}+\frac{15}{32}a+\frac{5}{24}$, $\frac{1}{3072}a^{9}+\frac{1}{512}a^{7}-\frac{5}{384}a^{6}-\frac{55}{3072}a^{5}-\frac{17}{384}a^{4}-\frac{23}{96}a^{3}+\frac{15}{64}a^{2}+\frac{7}{192}a-\frac{19}{48}$, $\frac{1}{73728}a^{10}-\frac{1}{8192}a^{9}+\frac{5}{36864}a^{8}-\frac{27}{4096}a^{7}+\frac{115}{24576}a^{6}+\frac{119}{73728}a^{5}-\frac{773}{18432}a^{4}-\frac{2047}{9216}a^{3}+\frac{1135}{4608}a^{2}+\frac{239}{4608}a+\frac{421}{1152}$, $\frac{1}{438019951951872}a^{11}-\frac{907094945}{146006650650624}a^{10}+\frac{9067956457}{109504987987968}a^{9}-\frac{24836684615}{73003325325312}a^{8}-\frac{506672895289}{146006650650624}a^{7}-\frac{6329006042707}{438019951951872}a^{6}+\frac{2216535372803}{219009975975936}a^{5}+\frac{42901020757}{27376246996992}a^{4}-\frac{2319334628251}{13688123498496}a^{3}-\frac{1515511855927}{27376246996992}a^{2}+\frac{1805368794239}{13688123498496}a+\frac{85245910287}{380225652736}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{16226}$, which has order $16226$ (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:   $\frac{684020699}{219009975975936}a^{11}+\frac{81030871}{24334441775104}a^{10}+\frac{8874575075}{54752493993984}a^{9}-\frac{1723716735}{12167220887552}a^{8}-\frac{101961908803}{73003325325312}a^{7}+\frac{2324828470975}{219009975975936}a^{6}-\frac{8093062725455}{109504987987968}a^{5}+\frac{2301503684231}{13688123498496}a^{4}+\frac{275779119703}{6844061749248}a^{3}+\frac{663356709427}{13688123498496}a^{2}+\frac{4189059228613}{6844061749248}a+\frac{114674340661}{190112826368}$, $\frac{279130199}{73003325325312}a^{11}+\frac{607953905}{219009975975936}a^{10}+\frac{1120398029}{6083610443776}a^{9}-\frac{26325366089}{109504987987968}a^{8}-\frac{60063569567}{24334441775104}a^{7}+\frac{1133428136939}{73003325325312}a^{6}-\frac{11170791212449}{109504987987968}a^{5}+\frac{3533200901681}{13688123498496}a^{4}+\frac{518682384529}{6844061749248}a^{3}+\frac{1551965285261}{13688123498496}a^{2}+\frac{6506713565003}{6844061749248}a+\frac{26820104380051}{1711015437312}$, $\frac{3395780903}{24334441775104}a^{11}-\frac{2301392765}{219009975975936}a^{10}+\frac{50316492367}{6083610443776}a^{9}-\frac{2651677748299}{109504987987968}a^{8}+\frac{882227149171}{24334441775104}a^{7}-\frac{18729626584751}{73003325325312}a^{6}-\frac{77050357531715}{109504987987968}a^{5}+\frac{22244079713179}{13688123498496}a^{4}+\frac{105668451235883}{6844061749248}a^{3}+\frac{278810098148311}{13688123498496}a^{2}+\frac{40678436515009}{6844061749248}a-\frac{7401105075463}{1711015437312}$, $\frac{28086780851}{219009975975936}a^{11}+\frac{1261960559}{24334441775104}a^{10}+\frac{386978209667}{54752493993984}a^{9}-\frac{189282909975}{12167220887552}a^{8}-\frac{1048669509403}{73003325325312}a^{7}+\frac{12932886227095}{219009975975936}a^{6}-\frac{190648971492791}{109504987987968}a^{5}+\frac{59258322193127}{13688123498496}a^{4}+\frac{64607220246151}{6844061749248}a^{3}+\frac{173665632200683}{13688123498496}a^{2}+\frac{109527185501053}{6844061749248}a+\frac{44150920795559}{570338479104}$, $\frac{458065319}{54752493993984}a^{11}-\frac{90638111}{18250831331328}a^{10}+\frac{7690712465}{13688123498496}a^{9}-\frac{18172982729}{9125415665664}a^{8}+\frac{45055187275}{6083610443776}a^{7}-\frac{2455175826989}{54752493993984}a^{6}+\frac{1909427694361}{27376246996992}a^{5}-\frac{753660163507}{3422030874624}a^{4}+\frac{2264109291697}{1711015437312}a^{3}+\frac{5818931853607}{3422030874624}a^{2}-\frac{1427853483299}{1711015437312}a-\frac{223540680385}{142584619776}$ 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:  \( 589381.2229376945 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 589381.2229376945 \cdot 16226}{2\cdot\sqrt{246168290724753398765625}}\cr\approx \mathstrut & 592.981202749421 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 62*x^10 - 226*x^9 + 577*x^8 - 2377*x^7 - 3136*x^6 + 12956*x^5 + 79168*x^4 + 61904*x^3 + 194304*x^2 + 378688*x + 235264)
 
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 + 62*x^10 - 226*x^9 + 577*x^8 - 2377*x^7 - 3136*x^6 + 12956*x^5 + 79168*x^4 + 61904*x^3 + 194304*x^2 + 378688*x + 235264, 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 + 62*x^10 - 226*x^9 + 577*x^8 - 2377*x^7 - 3136*x^6 + 12956*x^5 + 79168*x^4 + 61904*x^3 + 194304*x^2 + 378688*x + 235264);
 
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 + 62*x^10 - 226*x^9 + 577*x^8 - 2377*x^7 - 3136*x^6 + 12956*x^5 + 79168*x^4 + 61904*x^3 + 194304*x^2 + 378688*x + 235264);
 
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

\(\Q(\sqrt{-215}) \), \(\Q(\sqrt{129}) \), \(\Q(\sqrt{-15}) \), 3.3.1849.1, \(\Q(\sqrt{-15}, \sqrt{129})\), 6.0.18376055375.1, 6.6.3969227961.1, 6.0.11538453375.2

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 ${\href{/padicField/2.1.0.1}{1} }^{12}$ R R ${\href{/padicField/7.6.0.1}{6} }^{2}$ ${\href{/padicField/11.2.0.1}{2} }^{6}$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.6.0.1}{6} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{6}$ R ${\href{/padicField/47.2.0.1}{2} }^{6}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{6}$

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 3.6.3.1$x^{6} + 18 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} + 18 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(5\) Copy content Toggle raw display 5.6.3.2$x^{6} + 75 x^{2} - 375$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} + 75 x^{2} - 375$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(43\) Copy content Toggle raw display 43.12.10.1$x^{12} + 252 x^{11} + 26478 x^{10} + 1485540 x^{9} + 46993095 x^{8} + 797505912 x^{7} + 5770513850 x^{6} + 2392528572 x^{5} + 424071765 x^{4} + 103500180 x^{3} + 1995546888 x^{2} + 33432166152 x + 233626056556$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$