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Properties

Label 24.0.61138259958...0625.1
Degree $24$
Signature $[0, 12]$
Discriminant $5^{12}\cdot 7^{20}\cdot 11^{12}$
Root discriminant $37.53$
Ramified primes $5, 7, 11$
Class number $18$ (GRH)
Class group $[3, 6]$ (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, -3145728, 0, -458752, 0, 282624, 0, -24320, 0, -13104, 0, 3977, 0, -819, 0, -95, 0, 69, 0, -7, 0, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 3*x^22 - 7*x^20 + 69*x^18 - 95*x^16 - 819*x^14 + 3977*x^12 - 13104*x^10 - 24320*x^8 + 282624*x^6 - 458752*x^4 - 3145728*x^2 + 16777216)
 
gp: K = bnfinit(x^24 - 3*x^22 - 7*x^20 + 69*x^18 - 95*x^16 - 819*x^14 + 3977*x^12 - 13104*x^10 - 24320*x^8 + 282624*x^6 - 458752*x^4 - 3145728*x^2 + 16777216, 1)
 

Normalized defining polynomial

\( x^{24} - 3 x^{22} - 7 x^{20} + 69 x^{18} - 95 x^{16} - 819 x^{14} + 3977 x^{12} - 13104 x^{10} - 24320 x^{8} + 282624 x^{6} - 458752 x^{4} - 3145728 x^{2} + 16777216 \)

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:  \(61138259958814499261991852715087890625=5^{12}\cdot 7^{20}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.53$
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:  $5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(385=5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{385}(384,·)$, $\chi_{385}(1,·)$, $\chi_{385}(131,·)$, $\chi_{385}(199,·)$, $\chi_{385}(331,·)$, $\chi_{385}(76,·)$, $\chi_{385}(144,·)$, $\chi_{385}(274,·)$, $\chi_{385}(276,·)$, $\chi_{385}(89,·)$, $\chi_{385}(219,·)$, $\chi_{385}(221,·)$, $\chi_{385}(351,·)$, $\chi_{385}(34,·)$, $\chi_{385}(164,·)$, $\chi_{385}(166,·)$, $\chi_{385}(296,·)$, $\chi_{385}(109,·)$, $\chi_{385}(111,·)$, $\chi_{385}(241,·)$, $\chi_{385}(309,·)$, $\chi_{385}(54,·)$, $\chi_{385}(186,·)$, $\chi_{385}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{63632} a^{14} - \frac{3}{16} a^{12} + \frac{1}{16} a^{10} - \frac{3}{16} a^{8} - \frac{7}{16} a^{6} - \frac{1}{2} a^{5} + \frac{5}{16} a^{4} + \frac{1}{16} a^{2} - \frac{819}{3977}$, $\frac{1}{254528} a^{15} + \frac{5}{64} a^{13} + \frac{1}{64} a^{11} + \frac{13}{64} a^{9} + \frac{9}{64} a^{7} + \frac{21}{64} a^{5} - \frac{1}{2} a^{4} - \frac{15}{64} a^{3} - \frac{1199}{3977} a$, $\frac{1}{1018112} a^{16} - \frac{3}{1018112} a^{14} - \frac{15}{256} a^{12} - \frac{35}{256} a^{10} - \frac{39}{256} a^{8} + \frac{37}{256} a^{6} + \frac{1}{256} a^{4} - \frac{819}{63632} a^{2} - \frac{1}{2} a - \frac{95}{3977}$, $\frac{1}{4072448} a^{17} - \frac{3}{4072448} a^{15} - \frac{15}{1024} a^{13} + \frac{221}{1024} a^{11} + \frac{89}{1024} a^{9} - \frac{219}{1024} a^{7} - \frac{1}{2} a^{6} + \frac{257}{1024} a^{5} + \frac{62813}{254528} a^{3} - \frac{1}{2} a^{2} + \frac{1941}{7954} a$, $\frac{1}{32579584} a^{18} - \frac{1}{8144896} a^{17} - \frac{3}{32579584} a^{16} + \frac{3}{8144896} a^{15} - \frac{7}{32579584} a^{14} - \frac{241}{2048} a^{13} - \frac{35}{8192} a^{12} + \frac{35}{2048} a^{11} + \frac{345}{8192} a^{10} - \frac{345}{2048} a^{9} - \frac{475}{8192} a^{8} + \frac{475}{2048} a^{7} + \frac{1}{8192} a^{6} - \frac{1}{2048} a^{5} - \frac{819}{2036224} a^{4} - \frac{253709}{509056} a^{3} - \frac{95}{127264} a^{2} - \frac{15813}{31816} a + \frac{69}{7954}$, $\frac{1}{130318336} a^{19} + \frac{13}{130318336} a^{17} - \frac{55}{130318336} a^{15} + \frac{3821}{32768} a^{13} - \frac{4311}{32768} a^{11} + \frac{949}{32768} a^{9} + \frac{593}{32768} a^{7} - \frac{1016533}{4072448} a^{5} - \frac{64089}{254528} a^{3} - \frac{1}{2} a^{2} - \frac{1995}{7954} a - \frac{1}{2}$, $\frac{1}{521273344} a^{20} - \frac{3}{521273344} a^{18} - \frac{1}{8144896} a^{17} - \frac{7}{521273344} a^{16} + \frac{3}{8144896} a^{15} + \frac{69}{521273344} a^{14} + \frac{15}{2048} a^{13} - \frac{28327}{131072} a^{12} + \frac{291}{2048} a^{11} + \frac{20005}{131072} a^{10} - \frac{89}{2048} a^{9} + \frac{1}{131072} a^{8} - \frac{293}{2048} a^{7} - \frac{819}{32579584} a^{6} - \frac{769}{2048} a^{5} + \frac{1018017}{2036224} a^{4} - \frac{190077}{509056} a^{3} + \frac{69}{127264} a^{2} + \frac{509}{3977} a - \frac{7}{7954}$, $\frac{1}{2085093376} a^{21} - \frac{3}{2085093376} a^{19} + \frac{249}{2085093376} a^{17} - \frac{699}{2085093376} a^{15} - \frac{1}{127264} a^{14} - \frac{32167}{524288} a^{13} - \frac{5}{32} a^{12} - \frac{120027}{524288} a^{11} - \frac{1}{32} a^{10} + \frac{88321}{524288} a^{9} + \frac{3}{32} a^{8} - \frac{30226019}{130318336} a^{7} + \frac{7}{32} a^{6} - \frac{2034283}{4072448} a^{5} + \frac{11}{32} a^{4} + \frac{126889}{254528} a^{3} - \frac{1}{32} a^{2} - \frac{51}{15908} a + \frac{819}{7954}$, $\frac{1}{8340373504} a^{22} - \frac{3}{8340373504} a^{20} - \frac{7}{8340373504} a^{18} - \frac{1}{8144896} a^{17} + \frac{69}{8340373504} a^{16} - \frac{13}{8144896} a^{15} - \frac{95}{8340373504} a^{14} - \frac{65}{2048} a^{13} + \frac{151077}{2097152} a^{12} - \frac{237}{2048} a^{11} + \frac{1}{2097152} a^{10} - \frac{297}{2048} a^{9} - \frac{819}{521273344} a^{8} - \frac{437}{2048} a^{7} + \frac{16289697}{32579584} a^{6} + \frac{943}{2048} a^{5} - \frac{1018043}{2036224} a^{4} + \frac{62053}{254528} a^{3} + \frac{63625}{127264} a^{2} + \frac{2217}{7954} a - \frac{3}{7954}$, $\frac{1}{33361494016} a^{23} - \frac{3}{33361494016} a^{21} - \frac{7}{33361494016} a^{19} - \frac{4027}{33361494016} a^{17} - \frac{1}{2036224} a^{16} + \frac{12193}{33361494016} a^{15} + \frac{3}{2036224} a^{14} + \frac{212517}{8388608} a^{13} + \frac{15}{512} a^{12} - \frac{1953791}{8388608} a^{11} + \frac{35}{512} a^{10} + \frac{170023885}{2085093376} a^{9} - \frac{89}{512} a^{8} - \frac{2354479}{130318336} a^{7} - \frac{37}{512} a^{6} + \frac{508079}{2036224} a^{5} - \frac{1}{512} a^{4} - \frac{31613}{127264} a^{3} - \frac{62813}{127264} a^{2} + \frac{4023}{15908} a + \frac{95}{7954}$

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

Class group and class number

$C_{3}\times C_{6}$, which has order $18$ (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:  \( -\frac{7}{16289792} a^{20} - \frac{182685}{16289792} a^{6} \) (order $14$)
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:  \( 268997143.93463415 \) (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

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{77}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{385}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-35}, \sqrt{-55})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-11}, \sqrt{-35})\), \(\Q(\sqrt{-7}, \sqrt{-11})\), \(\Q(\sqrt{5}, \sqrt{77})\), \(\Q(\sqrt{-7}, \sqrt{-55})\), \(\Q(\sqrt{5}, \sqrt{-11})\), 6.0.2100875.1, 6.6.22370117.1, 6.0.399466375.2, \(\Q(\zeta_{7})\), 6.6.300125.1, 6.0.3195731.1, 6.6.2796264625.1, 8.0.21970650625.1, 12.0.7819095853026390625.1, 12.0.4413675765625.1, 12.0.7819095853026390625.3, 12.0.500422134593689.1, 12.12.7819095853026390625.1, 12.0.7819095853026390625.2, 12.0.159573384755640625.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{4}$ R R R ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
$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}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$