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Properties

Label 24.0.72340856237...0000.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{24}\cdot 3^{12}\cdot 5^{12}\cdot 7^{16}$
Root discriminant $28.34$
Ramified primes $2, 3, 5, 7$
Class number $3$ (GRH)
Class group $[3]$ (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![1, 0, -18, 0, 253, 0, -1062, 0, 3030, 0, -5271, 0, 6639, 0, -5124, 0, 2798, 0, -789, 0, 158, 0, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 15*x^22 + 158*x^20 - 789*x^18 + 2798*x^16 - 5124*x^14 + 6639*x^12 - 5271*x^10 + 3030*x^8 - 1062*x^6 + 253*x^4 - 18*x^2 + 1)
 
gp: K = bnfinit(x^24 - 15*x^22 + 158*x^20 - 789*x^18 + 2798*x^16 - 5124*x^14 + 6639*x^12 - 5271*x^10 + 3030*x^8 - 1062*x^6 + 253*x^4 - 18*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{24} - 15 x^{22} + 158 x^{20} - 789 x^{18} + 2798 x^{16} - 5124 x^{14} + 6639 x^{12} - 5271 x^{10} + 3030 x^{8} - 1062 x^{6} + 253 x^{4} - 18 x^{2} + 1 \)

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:  \(72340856237421875367936000000000000=2^{24}\cdot 3^{12}\cdot 5^{12}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.34$
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$$\chi_{420}(1,·)$, $\chi_{420}(331,·)$, $\chi_{420}(389,·)$, $\chi_{420}(71,·)$, $\chi_{420}(11,·)$, $\chi_{420}(79,·)$, $\chi_{420}(401,·)$, $\chi_{420}(211,·)$, $\chi_{420}(149,·)$, $\chi_{420}(151,·)$, $\chi_{420}(281,·)$, $\chi_{420}(239,·)$, $\chi_{420}(29,·)$, $\chi_{420}(289,·)$, $\chi_{420}(359,·)$, $\chi_{420}(169,·)$, $\chi_{420}(109,·)$, $\chi_{420}(221,·)$, $\chi_{420}(319,·)$, $\chi_{420}(179,·)$, $\chi_{420}(361,·)$, $\chi_{420}(121,·)$, $\chi_{420}(379,·)$, $\chi_{420}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{8} a^{18} - \frac{1}{4} a^{12} + \frac{1}{8} a^{6} - \frac{3}{8}$, $\frac{1}{8} a^{19} - \frac{1}{4} a^{13} + \frac{1}{8} a^{7} - \frac{3}{8} a$, $\frac{1}{183352} a^{20} + \frac{6473}{183352} a^{18} - \frac{9019}{22919} a^{16} - \frac{39485}{91676} a^{14} + \frac{39683}{91676} a^{12} - \frac{5605}{22919} a^{10} + \frac{26393}{183352} a^{8} - \frac{64591}{183352} a^{6} - \frac{88}{22919} a^{4} - \frac{31307}{183352} a^{2} + \frac{39581}{183352}$, $\frac{1}{183352} a^{21} + \frac{6473}{183352} a^{19} - \frac{9019}{22919} a^{17} - \frac{39485}{91676} a^{15} + \frac{39683}{91676} a^{13} - \frac{5605}{22919} a^{11} + \frac{26393}{183352} a^{9} - \frac{64591}{183352} a^{7} - \frac{88}{22919} a^{5} - \frac{31307}{183352} a^{3} + \frac{39581}{183352} a$, $\frac{1}{924825837832} a^{22} - \frac{298179}{924825837832} a^{20} - \frac{18273521245}{924825837832} a^{18} - \frac{14633772693}{462412918916} a^{16} - \frac{143736184233}{462412918916} a^{14} + \frac{4144244749}{35570224532} a^{12} - \frac{98160812223}{924825837832} a^{10} - \frac{322101635347}{924825837832} a^{8} - \frac{105247252133}{924825837832} a^{6} + \frac{46938583917}{924825837832} a^{4} - \frac{29822725259}{71140449064} a^{2} + \frac{272511138511}{924825837832}$, $\frac{1}{924825837832} a^{23} - \frac{298179}{924825837832} a^{21} - \frac{18273521245}{924825837832} a^{19} - \frac{14633772693}{462412918916} a^{17} - \frac{143736184233}{462412918916} a^{15} + \frac{4144244749}{35570224532} a^{13} - \frac{98160812223}{924825837832} a^{11} - \frac{322101635347}{924825837832} a^{9} - \frac{105247252133}{924825837832} a^{7} + \frac{46938583917}{924825837832} a^{5} - \frac{29822725259}{71140449064} a^{3} + \frac{272511138511}{924825837832} a$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( -\frac{148787121575}{924825837832} a^{23} + \frac{84389196267}{35570224532} a^{21} - \frac{22945059625745}{924825837832} a^{19} + \frac{55734503392911}{462412918916} a^{17} - \frac{96723380662449}{231206459458} a^{15} + \frac{329284836562241}{462412918916} a^{13} - \frac{61570749574763}{71140449064} a^{11} + \frac{272297101425195}{462412918916} a^{9} - \frac{264636377210105}{924825837832} a^{7} + \frac{54766886055237}{924825837832} a^{5} - \frac{1966812984841}{462412918916} a^{3} - \frac{3161652599205}{924825837832} a \) (order $12$)
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:  \( 39799331.07802784 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{15}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{15})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 6.0.153664.1, 6.6.4148928.1, 6.0.64827.1, 6.0.19208000.1, 6.6.300125.1, 6.0.8103375.1, 6.6.518616000.1, 8.0.12960000.1, 12.0.17213603549184.1, 12.0.368947264000000.1, 12.0.268962555456000000.3, 12.0.268962555456000000.1, 12.12.268962555456000000.1, 12.0.268962555456000000.2, 12.0.65664686390625.1

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$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$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$