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Properties

Label 24.0.17369039582...0000.3
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 $104$ (GRH)
Class group $[2, 52]$ (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![16777216, 0, 7340032, 0, 2162688, 0, 487424, 0, 78080, 0, 3696, 0, -3263, 0, 231, 0, 305, 0, 119, 0, 33, 0, 7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 7*x^22 + 33*x^20 + 119*x^18 + 305*x^16 + 231*x^14 - 3263*x^12 + 3696*x^10 + 78080*x^8 + 487424*x^6 + 2162688*x^4 + 7340032*x^2 + 16777216)
 
gp: K = bnfinit(x^24 + 7*x^22 + 33*x^20 + 119*x^18 + 305*x^16 + 231*x^14 - 3263*x^12 + 3696*x^10 + 78080*x^8 + 487424*x^6 + 2162688*x^4 + 7340032*x^2 + 16777216, 1)
 

Normalized defining polynomial

\( x^{24} + 7 x^{22} + 33 x^{20} + 119 x^{18} + 305 x^{16} + 231 x^{14} - 3263 x^{12} + 3696 x^{10} + 78080 x^{8} + 487424 x^{6} + 2162688 x^{4} + 7340032 x^{2} + 16777216 \)

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$$\chi_{420}(1,·)$, $\chi_{420}(389,·)$, $\chi_{420}(391,·)$, $\chi_{420}(331,·)$, $\chi_{420}(269,·)$, $\chi_{420}(271,·)$, $\chi_{420}(209,·)$, $\chi_{420}(211,·)$, $\chi_{420}(149,·)$, $\chi_{420}(151,·)$, $\chi_{420}(89,·)$, $\chi_{420}(29,·)$, $\chi_{420}(31,·)$, $\chi_{420}(419,·)$, $\chi_{420}(359,·)$, $\chi_{420}(361,·)$, $\chi_{420}(299,·)$, $\chi_{420}(239,·)$, $\chi_{420}(241,·)$, $\chi_{420}(179,·)$, $\chi_{420}(181,·)$, $\chi_{420}(121,·)$, $\chi_{420}(59,·)$, $\chi_{420}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{52208} a^{14} - \frac{7}{16} a^{12} - \frac{1}{16} a^{10} - \frac{7}{16} a^{8} - \frac{1}{16} a^{6} - \frac{7}{16} a^{4} - \frac{1}{16} a^{2} + \frac{231}{3263}$, $\frac{1}{208832} a^{15} - \frac{7}{64} a^{13} + \frac{31}{64} a^{11} + \frac{9}{64} a^{9} + \frac{15}{64} a^{7} + \frac{25}{64} a^{5} - \frac{1}{64} a^{3} + \frac{231}{13052} a$, $\frac{1}{835328} a^{16} + \frac{7}{835328} a^{14} + \frac{31}{256} a^{12} + \frac{73}{256} a^{10} + \frac{15}{256} a^{8} - \frac{39}{256} a^{6} - \frac{1}{256} a^{4} + \frac{231}{52208} a^{2} + \frac{305}{3263}$, $\frac{1}{3341312} a^{17} + \frac{7}{3341312} a^{15} + \frac{31}{1024} a^{13} + \frac{329}{1024} a^{11} - \frac{241}{1024} a^{9} + \frac{217}{1024} a^{7} + \frac{255}{1024} a^{5} - \frac{51977}{208832} a^{3} + \frac{892}{3263} a$, $\frac{1}{13365248} a^{18} + \frac{7}{13365248} a^{16} + \frac{33}{13365248} a^{14} + \frac{585}{4096} a^{12} - \frac{497}{4096} a^{10} - \frac{551}{4096} a^{8} - \frac{1}{4096} a^{6} + \frac{231}{835328} a^{4} + \frac{305}{52208} a^{2} + \frac{119}{3263}$, $\frac{1}{53460992} a^{19} + \frac{7}{53460992} a^{17} + \frac{33}{53460992} a^{15} + \frac{585}{16384} a^{13} + \frac{7695}{16384} a^{11} - \frac{4647}{16384} a^{9} + \frac{8191}{16384} a^{7} - \frac{1670425}{3341312} a^{5} - \frac{104111}{208832} a^{3} - \frac{6407}{13052} a$, $\frac{1}{213843968} a^{20} + \frac{7}{213843968} a^{18} + \frac{33}{213843968} a^{16} + \frac{119}{213843968} a^{14} + \frac{15887}{65536} a^{12} - \frac{29223}{65536} a^{10} - \frac{1}{65536} a^{8} + \frac{231}{13365248} a^{6} + \frac{305}{835328} a^{4} + \frac{119}{52208} a^{2} + \frac{33}{3263}$, $\frac{1}{855375872} a^{21} + \frac{7}{855375872} a^{19} + \frac{33}{855375872} a^{17} + \frac{119}{855375872} a^{15} + \frac{15887}{262144} a^{13} - \frac{29223}{262144} a^{11} + \frac{65535}{262144} a^{9} - \frac{13365017}{53460992} a^{7} + \frac{835633}{3341312} a^{5} - \frac{52089}{208832} a^{3} + \frac{824}{3263} a$, $\frac{1}{3421503488} a^{22} + \frac{7}{3421503488} a^{20} + \frac{33}{3421503488} a^{18} + \frac{119}{3421503488} a^{16} + \frac{305}{3421503488} a^{14} + \frac{36313}{1048576} a^{12} - \frac{1}{1048576} a^{10} + \frac{231}{213843968} a^{8} + \frac{305}{13365248} a^{6} + \frac{119}{835328} a^{4} + \frac{33}{52208} a^{2} + \frac{7}{3263}$, $\frac{1}{13686013952} a^{23} + \frac{7}{13686013952} a^{21} + \frac{33}{13686013952} a^{19} + \frac{119}{13686013952} a^{17} + \frac{305}{13686013952} a^{15} + \frac{36313}{4194304} a^{13} + \frac{1048575}{4194304} a^{11} - \frac{213843737}{855375872} a^{9} + \frac{13365553}{53460992} a^{7} - \frac{835209}{3341312} a^{5} + \frac{52241}{208832} a^{3} - \frac{814}{3263} a$

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

Class group and class number

$C_{2}\times C_{52}$, which has order $104$ (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:  \( \frac{171}{855375872} a^{23} + \frac{3859123}{855375872} a^{9} \) (order $28$)
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:  \( 233161229.71468583 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{7})^+\), \(\Q(i, \sqrt{105})\), \(\Q(i, \sqrt{7})\), \(\Q(i, \sqrt{15})\), \(\Q(\sqrt{7}, \sqrt{15})\), \(\Q(\sqrt{-7}, \sqrt{-15})\), \(\Q(\sqrt{7}, \sqrt{-15})\), \(\Q(\sqrt{-7}, \sqrt{15})\), 6.0.153664.1, 6.6.56723625.1, 6.0.3630312000.2, \(\Q(\zeta_{28})^+\), \(\Q(\zeta_{7})\), 6.6.518616000.1, 6.0.8103375.1, 8.0.31116960000.7, 12.0.13179165217344000000.2, \(\Q(\zeta_{28})\), 12.0.268962555456000000.3, 12.12.13179165217344000000.1, 12.0.3217569633140625.3, 12.0.13179165217344000000.10, 12.0.13179165217344000000.7

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$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$7.12.10.1$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}$