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Properties

Label 24.0.17369039582...0000.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{24}\cdot 3^{12}\cdot 5^{12}\cdot 7^{20}$
Root discriminant $39.20$
Ramified primes $2, 3, 5, 7$
Class number $312$ (GRH)
Class group $[2, 156]$ (GRH)
Galois group $C_2^2\times C_6$ (as 24T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![177241, 0, -54588, 0, -71174, 0, -25424, 0, -3091, 0, 24935, 0, 35678, 0, 23396, 0, 8972, 0, 2149, 0, 319, 0, 27, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 27*x^22 + 319*x^20 + 2149*x^18 + 8972*x^16 + 23396*x^14 + 35678*x^12 + 24935*x^10 - 3091*x^8 - 25424*x^6 - 71174*x^4 - 54588*x^2 + 177241)
 
gp: K = bnfinit(x^24 + 27*x^22 + 319*x^20 + 2149*x^18 + 8972*x^16 + 23396*x^14 + 35678*x^12 + 24935*x^10 - 3091*x^8 - 25424*x^6 - 71174*x^4 - 54588*x^2 + 177241, 1)
 

Normalized defining polynomial

\( x^{24} + 27 x^{22} + 319 x^{20} + 2149 x^{18} + 8972 x^{16} + 23396 x^{14} + 35678 x^{12} + 24935 x^{10} - 3091 x^{8} - 25424 x^{6} - 71174 x^{4} - 54588 x^{2} + 177241 \)

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

Invariants

Degree:  $24$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 12]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(173690395826049922758414336000000000000=2^{24}\cdot 3^{12}\cdot 5^{12}\cdot 7^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.20$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(420=2^{2}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{420}(1,·)$, $\chi_{420}(389,·)$, $\chi_{420}(341,·)$, $\chi_{420}(391,·)$, $\chi_{420}(11,·)$, $\chi_{420}(271,·)$, $\chi_{420}(149,·)$, $\chi_{420}(409,·)$, $\chi_{420}(71,·)$, $\chi_{420}(79,·)$, $\chi_{420}(29,·)$, $\chi_{420}(31,·)$, $\chi_{420}(419,·)$, $\chi_{420}(101,·)$, $\chi_{420}(379,·)$, $\chi_{420}(41,·)$, $\chi_{420}(299,·)$, $\chi_{420}(191,·)$, $\chi_{420}(349,·)$, $\chi_{420}(229,·)$, $\chi_{420}(361,·)$, $\chi_{420}(121,·)$, $\chi_{420}(59,·)$, $\chi_{420}(319,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{377} a^{14} + \frac{14}{377} a^{12} + \frac{77}{377} a^{10} - \frac{167}{377} a^{8} - \frac{83}{377} a^{6} - \frac{181}{377} a^{4} + \frac{49}{377} a^{2} + \frac{146}{377}$, $\frac{1}{377} a^{15} + \frac{14}{377} a^{13} + \frac{77}{377} a^{11} - \frac{167}{377} a^{9} - \frac{83}{377} a^{7} - \frac{181}{377} a^{5} + \frac{49}{377} a^{3} + \frac{146}{377} a$, $\frac{1}{377} a^{16} - \frac{119}{377} a^{12} - \frac{114}{377} a^{10} - \frac{7}{377} a^{8} - \frac{150}{377} a^{6} - \frac{56}{377} a^{4} - \frac{163}{377} a^{2} - \frac{159}{377}$, $\frac{1}{377} a^{17} - \frac{119}{377} a^{13} - \frac{114}{377} a^{11} - \frac{7}{377} a^{9} - \frac{150}{377} a^{7} - \frac{56}{377} a^{5} - \frac{163}{377} a^{3} - \frac{159}{377} a$, $\frac{1}{3016} a^{18} + \frac{1}{1508} a^{16} - \frac{1}{3016} a^{14} - \frac{103}{232} a^{12} - \frac{197}{3016} a^{10} + \frac{61}{377} a^{8} - \frac{25}{104} a^{6} - \frac{1275}{3016} a^{4} + \frac{575}{1508} a^{2} + \frac{1453}{3016}$, $\frac{1}{1269736} a^{19} - \frac{571}{634868} a^{17} + \frac{1343}{1269736} a^{15} - \frac{21743}{43784} a^{13} + \frac{532291}{1269736} a^{11} + \frac{43128}{158717} a^{9} + \frac{228219}{1269736} a^{7} + \frac{329229}{1269736} a^{5} - \frac{153749}{634868} a^{3} + \frac{295125}{1269736} a$, $\frac{1}{267914296} a^{20} - \frac{5773}{267914296} a^{18} - \frac{209999}{267914296} a^{16} + \frac{23709}{66978574} a^{14} + \frac{2728151}{33489287} a^{12} - \frac{34483885}{267914296} a^{10} - \frac{107015637}{267914296} a^{8} - \frac{6243587}{33489287} a^{6} + \frac{130246707}{267914296} a^{4} + \frac{122902955}{267914296} a^{2} + \frac{114249}{636376}$, $\frac{1}{267914296} a^{21} - \frac{19}{66978574} a^{19} - \frac{320141}{267914296} a^{17} - \frac{71221}{267914296} a^{15} + \frac{113598181}{267914296} a^{13} + \frac{29724231}{133957148} a^{11} - \frac{95695909}{267914296} a^{9} - \frac{131995413}{267914296} a^{7} + \frac{498020}{1154803} a^{5} - \frac{107415783}{267914296} a^{3} + \frac{53508285}{133957148} a$, $\frac{1}{267914296} a^{22} + \frac{20295}{133957148} a^{18} - \frac{219227}{267914296} a^{16} - \frac{46637}{133957148} a^{14} - \frac{45930559}{267914296} a^{12} - \frac{19569825}{66978574} a^{10} + \frac{25946391}{267914296} a^{8} + \frac{93247117}{267914296} a^{6} - \frac{3050340}{33489287} a^{4} + \frac{95427}{1154803} a^{2} + \frac{61675}{636376}$, $\frac{1}{267914296} a^{23} + \frac{3}{10304396} a^{19} - \frac{146643}{267914296} a^{17} + \frac{109503}{133957148} a^{15} - \frac{120070895}{267914296} a^{13} - \frac{26743825}{66978574} a^{11} + \frac{6171471}{267914296} a^{9} - \frac{67436979}{267914296} a^{7} - \frac{6283704}{33489287} a^{5} - \frac{10328332}{33489287} a^{3} - \frac{79372777}{267914296} a$

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

Class group and class number

$C_{2}\times C_{156}$, which has order $312$ (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:  $11$
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:  \( 4345227.172827557 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_6$ (as 24T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 24
The 24 conjugacy class representatives for $C_2^2\times C_6$
Character table for $C_2^2\times C_6$ is not computed

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{3}, \sqrt{-35})\), \(\Q(\sqrt{-5}, \sqrt{7})\), \(\Q(\sqrt{-15}, \sqrt{21})\), \(\Q(\sqrt{-5}, \sqrt{21})\), \(\Q(\sqrt{7}, \sqrt{-15})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{3}, \sqrt{7})\), 6.0.2100875.1, 6.0.3630312000.2, 6.6.4148928.1, 6.0.19208000.1, \(\Q(\zeta_{28})^+\), \(\Q(\zeta_{21})^+\), 6.0.8103375.1, 8.0.31116960000.5, 12.0.13179165217344000000.9, 12.0.18078415936000000.1, 12.0.3217569633140625.1, 12.0.13179165217344000000.3, 12.0.13179165217344000000.10, 12.0.268962555456000000.1, \(\Q(\zeta_{84})^+\)

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

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 R ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$