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Properties

Label 32.0.67428764606...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{64}\cdot 5^{24}\cdot 23^{16}$
Root discriminant $64.14$
Ramified primes $2, 5, 23$
Class number Not computed
Class group Not computed
Galois group $C_2^3\times C_4$ (as 32T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2821109907456, 0, 0, 0, -106662334464, 0, 0, 0, 1855975680, 0, 0, 0, 12129264, 0, 0, 0, -1890671, 0, 0, 0, 9359, 0, 0, 0, 1105, 0, 0, 0, -49, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 49*x^28 + 1105*x^24 + 9359*x^20 - 1890671*x^16 + 12129264*x^12 + 1855975680*x^8 - 106662334464*x^4 + 2821109907456)
 
gp: K = bnfinit(x^32 - 49*x^28 + 1105*x^24 + 9359*x^20 - 1890671*x^16 + 12129264*x^12 + 1855975680*x^8 - 106662334464*x^4 + 2821109907456, 1)
 

Normalized defining polynomial

\( x^{32} - 49 x^{28} + 1105 x^{24} + 9359 x^{20} - 1890671 x^{16} + 12129264 x^{12} + 1855975680 x^{8} - 106662334464 x^{4} + 2821109907456 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6742876460661466820072365921140736000000000000000000000000=2^{64}\cdot 5^{24}\cdot 23^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.14$
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:  $2, 5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(920=2^{3}\cdot 5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{920}(1,·)$, $\chi_{920}(643,·)$, $\chi_{920}(137,·)$, $\chi_{920}(139,·)$, $\chi_{920}(781,·)$, $\chi_{920}(783,·)$, $\chi_{920}(277,·)$, $\chi_{920}(919,·)$, $\chi_{920}(413,·)$, $\chi_{920}(551,·)$, $\chi_{920}(553,·)$, $\chi_{920}(47,·)$, $\chi_{920}(689,·)$, $\chi_{920}(691,·)$, $\chi_{920}(183,·)$, $\chi_{920}(827,·)$, $\chi_{920}(829,·)$, $\chi_{920}(321,·)$, $\chi_{920}(323,·)$, $\chi_{920}(459,·)$, $\chi_{920}(461,·)$, $\chi_{920}(597,·)$, $\chi_{920}(599,·)$, $\chi_{920}(91,·)$, $\chi_{920}(93,·)$, $\chi_{920}(737,·)$, $\chi_{920}(229,·)$, $\chi_{920}(231,·)$, $\chi_{920}(873,·)$, $\chi_{920}(367,·)$, $\chi_{920}(369,·)$, $\chi_{920}(507,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{11} a^{16} + \frac{3}{11} a^{12} - \frac{2}{11} a^{8} + \frac{5}{11} a^{4} + \frac{4}{11}$, $\frac{1}{66} a^{17} - \frac{19}{66} a^{13} + \frac{31}{66} a^{9} + \frac{5}{66} a^{5} - \frac{29}{66} a$, $\frac{1}{396} a^{18} - \frac{85}{396} a^{14} + \frac{97}{396} a^{10} + \frac{71}{396} a^{6} - \frac{95}{396} a^{2}$, $\frac{1}{2376} a^{19} - \frac{481}{2376} a^{15} + \frac{889}{2376} a^{11} + \frac{71}{2376} a^{7} - \frac{887}{2376} a^{3}$, $\frac{1}{26953405776} a^{20} + \frac{289}{14256} a^{16} - \frac{1201}{14256} a^{12} - \frac{2063}{14256} a^{8} + \frac{3887}{14256} a^{4} - \frac{3771983}{20797381}$, $\frac{1}{161720434656} a^{21} + \frac{289}{85536} a^{17} + \frac{41567}{85536} a^{13} - \frac{16319}{85536} a^{9} - \frac{38881}{85536} a^{5} - \frac{45366745}{124784286} a$, $\frac{1}{970322607936} a^{22} + \frac{289}{513216} a^{18} - \frac{215041}{513216} a^{14} - \frac{101855}{513216} a^{10} - \frac{124417}{513216} a^{6} - \frac{294935317}{748705716} a^{2}$, $\frac{1}{5821935647616} a^{23} + \frac{289}{3079296} a^{19} + \frac{1324607}{3079296} a^{15} - \frac{615071}{3079296} a^{11} + \frac{902015}{3079296} a^{7} + \frac{453770399}{4492234296} a^{3}$, $\frac{1}{34931613885696} a^{24} - \frac{49}{34931613885696} a^{20} - \frac{43969}{18475776} a^{16} + \frac{1779937}{18475776} a^{12} + \frac{6718463}{18475776} a^{8} + \frac{2450318975}{26953405776} a^{4} + \frac{515738}{1890671}$, $\frac{1}{209589683314176} a^{25} - \frac{49}{209589683314176} a^{21} - \frac{43969}{110854656} a^{17} + \frac{1779937}{110854656} a^{13} - \frac{30233089}{110854656} a^{9} + \frac{29403724751}{161720434656} a^{5} - \frac{5156275}{11344026} a$, $\frac{1}{1257538099885056} a^{26} - \frac{49}{1257538099885056} a^{22} - \frac{43969}{665127936} a^{18} + \frac{223489249}{665127936} a^{14} - \frac{251942401}{665127936} a^{10} + \frac{191124159407}{970322607936} a^{6} + \frac{17531777}{68064156} a^{2}$, $\frac{1}{7545228599310336} a^{27} - \frac{49}{7545228599310336} a^{23} - \frac{43969}{3990767616} a^{19} - \frac{441638687}{3990767616} a^{15} + \frac{1743441407}{3990767616} a^{11} - \frac{779198448529}{5821935647616} a^{7} + \frac{153660089}{408384936} a^{3}$, $\frac{1}{45271371595862016} a^{28} - \frac{49}{45271371595862016} a^{24} + \frac{1105}{45271371595862016} a^{20} - \frac{754047263}{23944605696} a^{16} + \frac{4353564671}{23944605696} a^{12} - \frac{15878006302321}{34931613885696} a^{8} - \frac{9801237359}{26953405776} a^{4} - \frac{1890720}{20797381}$, $\frac{1}{271628229575172096} a^{29} - \frac{49}{271628229575172096} a^{25} + \frac{1105}{271628229575172096} a^{21} - \frac{754047263}{143667634176} a^{17} - \frac{67480252417}{143667634176} a^{13} - \frac{15878006302321}{209589683314176} a^{9} + \frac{71058979969}{161720434656} a^{5} - \frac{21742741}{62392143} a$, $\frac{1}{1629769377451032576} a^{30} - \frac{49}{1629769377451032576} a^{26} + \frac{1105}{1629769377451032576} a^{22} - \frac{754047263}{862005805056} a^{18} - \frac{211147886593}{862005805056} a^{14} - \frac{225467689616497}{1257538099885056} a^{10} + \frac{232779414625}{970322607936} a^{6} - \frac{146527027}{374352858} a^{2}$, $\frac{1}{9778616264706195456} a^{31} - \frac{49}{9778616264706195456} a^{27} + \frac{1105}{9778616264706195456} a^{23} - \frac{754047263}{5172034830336} a^{19} - \frac{211147886593}{5172034830336} a^{15} - \frac{2740543889386609}{7545228599310336} a^{11} - \frac{1707865801247}{5821935647616} a^{7} + \frac{227825831}{2246117148} a^{3}$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{365}{209589683314176} a^{29} - \frac{75943352179}{209589683314176} a^{9} \) (order $40$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3\times C_4$ (as 32T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2^3\times C_4$
Character table for $C_2^3\times C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-230}) \), \(\Q(\sqrt{230}) \), \(\Q(\sqrt{-115}) \), \(\Q(\sqrt{115}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-46}) \), \(\Q(\sqrt{46}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{23}) \), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{230})\), \(\Q(i, \sqrt{115})\), \(\Q(\sqrt{2}, \sqrt{-115})\), \(\Q(\sqrt{2}, \sqrt{115})\), \(\Q(\sqrt{-2}, \sqrt{115})\), \(\Q(\sqrt{-2}, \sqrt{-115})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{46})\), \(\Q(i, \sqrt{23})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{-23})\), \(\Q(\sqrt{2}, \sqrt{23})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{23})\), \(\Q(\sqrt{-2}, \sqrt{-23})\), \(\Q(\sqrt{5}, \sqrt{-46})\), \(\Q(\sqrt{-5}, \sqrt{46})\), \(\Q(\sqrt{10}, \sqrt{-23})\), \(\Q(\sqrt{-10}, \sqrt{23})\), \(\Q(\sqrt{5}, \sqrt{46})\), \(\Q(\sqrt{-5}, \sqrt{-46})\), \(\Q(\sqrt{10}, \sqrt{23})\), \(\Q(\sqrt{-10}, \sqrt{-23})\), \(\Q(\sqrt{5}, \sqrt{-23})\), \(\Q(\sqrt{-5}, \sqrt{23})\), \(\Q(\sqrt{10}, \sqrt{-46})\), \(\Q(\sqrt{-10}, \sqrt{46})\), \(\Q(\sqrt{5}, \sqrt{23})\), \(\Q(\sqrt{-5}, \sqrt{-23})\), \(\Q(\sqrt{10}, \sqrt{46})\), \(\Q(\sqrt{-10}, \sqrt{-46})\), \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), 4.0.8000.2, 4.4.8000.1, 4.4.4232000.1, 4.0.4232000.2, 4.4.66125.1, 4.0.1058000.1, 8.0.11462287360000.9, 8.0.40960000.1, 8.0.18339659776.1, 8.0.11462287360000.8, 8.0.11462287360000.4, 8.0.44774560000.1, 8.0.11462287360000.3, 8.0.716392960000.2, 8.0.11462287360000.2, 8.8.11462287360000.1, 8.0.11462287360000.1, 8.0.11462287360000.7, 8.0.11462287360000.5, 8.0.716392960000.1, 8.0.11462287360000.6, \(\Q(\zeta_{20})\), 8.0.1024000000.2, 8.0.286557184000000.28, 8.0.1119364000000.2, 8.0.64000000.2, \(\Q(\zeta_{40})^+\), 8.8.17909824000000.2, 8.0.286557184000000.40, 8.0.64000000.1, 8.0.1024000000.1, 8.0.286557184000000.38, 8.0.17909824000000.23, 8.0.17909824000000.14, 8.0.286557184000000.25, 8.0.17909824000000.10, 8.0.286557184000000.34, 8.0.17909824000000.15, 8.8.286557184000000.3, 8.0.286557184000000.32, 8.8.17909824000000.1, 8.0.4372515625.1, 8.0.1119364000000.1, 8.0.17909824000000.18, 8.0.17909824000000.12, 8.0.1119364000000.3, 8.8.1119364000000.1, 8.0.286557184000000.26, 8.8.286557184000000.2, 16.0.131384031523215769600000000.1, \(\Q(\zeta_{40})\), 16.0.82115019702009856000000000000.7, 16.0.82115019702009856000000000000.8, 16.0.82115019702009856000000000000.1, 16.0.1252975764496000000000000.1, 16.0.82115019702009856000000000000.2, 16.0.320761795710976000000000000.1, 16.0.82115019702009856000000000000.4, 16.0.82115019702009856000000000000.6, 16.16.82115019702009856000000000000.1, 16.0.82115019702009856000000000000.5, 16.0.82115019702009856000000000000.3, 16.0.320761795710976000000000000.2, 16.0.82115019702009856000000000000.9

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{32}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$23$23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$