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Properties

Label 12.0.19176887936...8000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{24}\cdot 3^{20}\cdot 5^{3}\cdot 11^{6}\cdot 23^{6}$
Root discriminant $593.70$
Ramified primes $2, 3, 5, 11, 23$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 12T205

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1130499648312, 293930060688, 65875535760, -15275047040, 361635132, 31301160, -438096, -12144, 19956, -3036, -24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 24*x^10 - 3036*x^9 + 19956*x^8 - 12144*x^7 - 438096*x^6 + 31301160*x^5 + 361635132*x^4 - 15275047040*x^3 + 65875535760*x^2 + 293930060688*x + 1130499648312)
 
gp: K = bnfinit(x^12 - 24*x^10 - 3036*x^9 + 19956*x^8 - 12144*x^7 - 438096*x^6 + 31301160*x^5 + 361635132*x^4 - 15275047040*x^3 + 65875535760*x^2 + 293930060688*x + 1130499648312, 1)
 

Normalized defining polynomial

\( x^{12} - 24 x^{10} - 3036 x^{9} + 19956 x^{8} - 12144 x^{7} - 438096 x^{6} + 31301160 x^{5} + 361635132 x^{4} - 15275047040 x^{3} + 65875535760 x^{2} + 293930060688 x + 1130499648312 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1917688793698627249096001323008000=2^{24}\cdot 3^{20}\cdot 5^{3}\cdot 11^{6}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $593.70$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5, 11, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{8971914624819237059536489162468922320803607885841401313884} a^{11} - \frac{148423157742568317136530791355234121763103016697299602101}{2242978656204809264884122290617230580200901971460350328471} a^{10} - \frac{213900860847397811041540883963503493323698270373088694753}{4485957312409618529768244581234461160401803942920700656942} a^{9} + \frac{390062142022017206221050328638759575177492287156952749513}{4485957312409618529768244581234461160401803942920700656942} a^{8} + \frac{1330382258577428204779090053745058522297153172372656500}{31591248678940975561748201276299022256350731992399300401} a^{7} + \frac{555038988370875211942815588546157461634855673445913723126}{2242978656204809264884122290617230580200901971460350328471} a^{6} + \frac{534488312591006645547461946153665960535577910273302533007}{2242978656204809264884122290617230580200901971460350328471} a^{5} + \frac{76098679635781555790689848055930261004704321500332074054}{2242978656204809264884122290617230580200901971460350328471} a^{4} + \frac{979361777864747934941635539838322600844445534376746090886}{2242978656204809264884122290617230580200901971460350328471} a^{3} - \frac{484565844641545652801556918575312938071063043397597818517}{2242978656204809264884122290617230580200901971460350328471} a^{2} - \frac{756425161221653571782557346460505258692276241125860859425}{2242978656204809264884122290617230580200901971460350328471} a + \frac{817836998399518472938166017181051121278794088791707963084}{2242978656204809264884122290617230580200901971460350328471}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $5$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 53240397144.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

12T205:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1152
The 24 conjugacy class representatives for [E(4)^3:3:2]3
Character table for [E(4)^3:3:2]3 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: data not computed
Degree 36 siblings: data not computed

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 ${\href{/LocalNumberField/7.12.0.1}{12} }$ R ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.12.24.50$x^{12} + 36 x^{11} - 48 x^{10} - 12 x^{9} - 58 x^{8} - 32 x^{6} + 48 x^{5} + 60 x^{4} - 16 x^{3} + 64 x^{2} - 48 x + 40$$4$$3$$24$12T205$[4/3, 4/3, 4/3, 4/3, 8/3, 8/3]_{3}^{6}$
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.9.16.12$x^{9} + 6 x^{8} + 3$$9$$1$$16$$S_3\times C_3$$[2, 2]^{2}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$11$11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$23$23.12.6.2$x^{12} - 6436343 x^{2} + 2220538335$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$