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Properties

Label 32.0.247...625.1
Degree $32$
Signature $[0, 16]$
Discriminant $2.479\times 10^{37}$
Root discriminant $14.74$
Ramified primes $5, 13$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_8.A_4$ (as 32T402)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 + x^29 + 14*x^28 - 23*x^27 - 25*x^26 + 77*x^25 + 32*x^24 - 233*x^23 + 130*x^22 + 275*x^21 - 204*x^20 - 500*x^19 + 369*x^18 + 976*x^17 - 775*x^16 - 1988*x^15 + 2706*x^14 + 1255*x^13 - 4074*x^12 + 870*x^11 + 3435*x^10 - 2784*x^9 - 528*x^8 + 1624*x^7 - 465*x^6 - 564*x^5 + 639*x^4 - 293*x^3 + 55*x^2 - x + 1)
 
gp: K = bnfinit(x^32 - 2*x^31 + x^29 + 14*x^28 - 23*x^27 - 25*x^26 + 77*x^25 + 32*x^24 - 233*x^23 + 130*x^22 + 275*x^21 - 204*x^20 - 500*x^19 + 369*x^18 + 976*x^17 - 775*x^16 - 1988*x^15 + 2706*x^14 + 1255*x^13 - 4074*x^12 + 870*x^11 + 3435*x^10 - 2784*x^9 - 528*x^8 + 1624*x^7 - 465*x^6 - 564*x^5 + 639*x^4 - 293*x^3 + 55*x^2 - x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 55, -293, 639, -564, -465, 1624, -528, -2784, 3435, 870, -4074, 1255, 2706, -1988, -775, 976, 369, -500, -204, 275, 130, -233, 32, 77, -25, -23, 14, 1, 0, -2, 1]);
 

\( x^{32} - 2 x^{31} + x^{29} + 14 x^{28} - 23 x^{27} - 25 x^{26} + 77 x^{25} + 32 x^{24} - 233 x^{23} + 130 x^{22} + 275 x^{21} - 204 x^{20} - 500 x^{19} + 369 x^{18} + 976 x^{17} - 775 x^{16} - 1988 x^{15} + 2706 x^{14} + 1255 x^{13} - 4074 x^{12} + 870 x^{11} + 3435 x^{10} - 2784 x^{9} - 528 x^{8} + 1624 x^{7} - 465 x^{6} - 564 x^{5} + 639 x^{4} - 293 x^{3} + 55 x^{2} - x + 1 \)

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

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(24788700386255228556692600250244140625\)\(\medspace = 5^{28}\cdot 13^{16}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $14.74$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $5, 13$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{421} a^{30} + \frac{189}{421} a^{29} - \frac{50}{421} a^{28} - \frac{39}{421} a^{27} - \frac{181}{421} a^{26} - \frac{190}{421} a^{25} + \frac{99}{421} a^{24} + \frac{157}{421} a^{23} - \frac{123}{421} a^{22} - \frac{42}{421} a^{21} - \frac{168}{421} a^{20} - \frac{106}{421} a^{19} - \frac{135}{421} a^{18} + \frac{90}{421} a^{17} + \frac{181}{421} a^{16} - \frac{160}{421} a^{15} + \frac{32}{421} a^{14} + \frac{56}{421} a^{13} + \frac{70}{421} a^{12} + \frac{135}{421} a^{11} + \frac{20}{421} a^{10} + \frac{176}{421} a^{9} - \frac{120}{421} a^{8} - \frac{95}{421} a^{7} + \frac{168}{421} a^{6} + \frac{90}{421} a^{5} + \frac{199}{421} a^{4} + \frac{54}{421} a^{3} - \frac{17}{421} a^{2} - \frac{41}{421} a + \frac{96}{421}$, $\frac{1}{517148361018374107192113327427772720512229} a^{31} + \frac{430805350340824651525469753882864583858}{517148361018374107192113327427772720512229} a^{30} + \frac{27627694179173548158188749248595791237297}{517148361018374107192113327427772720512229} a^{29} - \frac{464680664818468584713677567518084033525}{1228380905031767475515708616217987459649} a^{28} + \frac{248407000550493038860296954157724044116676}{517148361018374107192113327427772720512229} a^{27} + \frac{213646319340272871134728851276575306684770}{517148361018374107192113327427772720512229} a^{26} + \frac{236816491942312863094270540791292035323861}{517148361018374107192113327427772720512229} a^{25} + \frac{88629003397520214146681506232912337573262}{517148361018374107192113327427772720512229} a^{24} + \frac{56592518430908620553208482473168923740512}{517148361018374107192113327427772720512229} a^{23} - \frac{170916112055014706383135683723138111367709}{517148361018374107192113327427772720512229} a^{22} + \frac{65640843811658512135150859851821641013562}{517148361018374107192113327427772720512229} a^{21} - \frac{91667372340354381546958780722483177843643}{517148361018374107192113327427772720512229} a^{20} + \frac{1155472194241730407324151362755099682278}{4744480376315358781579021352548373582681} a^{19} + \frac{21623831458152184019315080147502879828542}{517148361018374107192113327427772720512229} a^{18} + \frac{258228635727781823356130024098507239135602}{517148361018374107192113327427772720512229} a^{17} + \frac{158456988986831857369639877204009037006739}{517148361018374107192113327427772720512229} a^{16} + \frac{213110684078708192853893471657007227185156}{517148361018374107192113327427772720512229} a^{15} - \frac{42953082617150918504834338800660710103876}{517148361018374107192113327427772720512229} a^{14} - \frac{219125144950476361959017609416559014443632}{517148361018374107192113327427772720512229} a^{13} - \frac{160138982699928664992754086762163920547472}{517148361018374107192113327427772720512229} a^{12} - \frac{118972979296600603548876684057577475077279}{517148361018374107192113327427772720512229} a^{11} - \frac{142556138803531238020395537403295034436812}{517148361018374107192113327427772720512229} a^{10} - \frac{172860824390552831355007990369548256002499}{517148361018374107192113327427772720512229} a^{9} + \frac{120592322082442828708132475676728730141012}{517148361018374107192113327427772720512229} a^{8} + \frac{160040757229481672434861189817883166384715}{517148361018374107192113327427772720512229} a^{7} + \frac{165191014833135764428376230972861896892163}{517148361018374107192113327427772720512229} a^{6} - \frac{180576967788478650186819853503579061920927}{517148361018374107192113327427772720512229} a^{5} - \frac{214007854917660186745641928509757730909213}{517148361018374107192113327427772720512229} a^{4} + \frac{35561032050330356703608901964878357616847}{517148361018374107192113327427772720512229} a^{3} - \frac{60170917769748571427345659557275634638}{517148361018374107192113327427772720512229} a^{2} + \frac{46336910681507966444588089948349167057813}{517148361018374107192113327427772720512229} a + \frac{141952077080397152566331110758560354541958}{517148361018374107192113327427772720512229}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{6423504503897763274593831789201417}{13582138582017456975740811356110849} a^{31} + \frac{7178582452064768517573258172256749}{13582138582017456975740811356110849} a^{30} + \frac{7067206241421796320356437201706777}{13582138582017456975740811356110849} a^{29} - \frac{1387658770165721849722628031402285}{13582138582017456975740811356110849} a^{28} - \frac{91687916282481827262138240324224439}{13582138582017456975740811356110849} a^{27} + \frac{67505959129681174008770660844121277}{13582138582017456975740811356110849} a^{26} + \frac{230816759281668032679765042160439415}{13582138582017456975740811356110849} a^{25} - \frac{303917237527655503407448652478540044}{13582138582017456975740811356110849} a^{24} - \frac{498333409341672780997534407394437763}{13582138582017456975740811356110849} a^{23} + \frac{1105613185044314640357390839560857893}{13582138582017456975740811356110849} a^{22} + \frac{185939608096098732574143187822123175}{13582138582017456975740811356110849} a^{21} - \frac{1761136541240886568117589419166090260}{13582138582017456975740811356110849} a^{20} - \frac{1975658090272192291210630468746397}{124606775981811531887530379413861} a^{19} + \frac{3250542113034486814507257709811605278}{13582138582017456975740811356110849} a^{18} + \frac{438925690650856033356595708791793706}{13582138582017456975740811356110849} a^{17} - \frac{6297357585930183706348102420950697120}{13582138582017456975740811356110849} a^{16} - \frac{472959949491997546585316064864329249}{13582138582017456975740811356110849} a^{15} + \frac{13162977646406243048540825239190696596}{13582138582017456975740811356110849} a^{14} - \frac{6016079853381329578031803061953102364}{13582138582017456975740811356110849} a^{13} - \frac{15029858183900166867855796226263769568}{13582138582017456975740811356110849} a^{12} + \frac{14248454465299087378515964043834627856}{13582138582017456975740811356110849} a^{11} + \frac{8617334512911968181073810535918181761}{13582138582017456975740811356110849} a^{10} - \frac{16888081068548984322439802448482682320}{13582138582017456975740811356110849} a^{9} + \frac{2468421839294359809519159076309269927}{13582138582017456975740811356110849} a^{8} + \frac{8022833735917993911655954850849516188}{13582138582017456975740811356110849} a^{7} - \frac{4285955043396325216690318456045600163}{13582138582017456975740811356110849} a^{6} - \frac{1687070555771401240397734324093192233}{13582138582017456975740811356110849} a^{5} + \frac{2913089983939363212288096549397022023}{13582138582017456975740811356110849} a^{4} - \frac{1487993229528095286120908123732764408}{13582138582017456975740811356110849} a^{3} + \frac{195403714016035807910534405552108474}{13582138582017456975740811356110849} a^{2} + \frac{97567936121317121148810131759033208}{13582138582017456975740811356110849} a + \frac{3023404456381992601855054844715558}{13582138582017456975740811356110849} \) (order $10$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1271231.1521675275 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 1271231.1521675275 \cdot 1}{10\sqrt{24788700386255228556692600250244140625}}\approx 0.150652221984326$ (assuming GRH)

Galois group

$C_8.A_4$ (as 32T402):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 96
The 28 conjugacy class representatives for $C_8.A_4$
Character table for $C_8.A_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.0.4225.1, 8.0.17850625.1, 16.0.199153008056640625.1

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 $24{,}\,{\href{/LocalNumberField/2.8.0.1}{8} }$ $24{,}\,{\href{/LocalNumberField/3.8.0.1}{8} }$ R $24{,}\,{\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R $24{,}\,{\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ $24{,}\,{\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ $24{,}\,{\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ $24{,}\,{\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
13Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.65.6t1.b.a$1$ $ 5 \cdot 13 $ 6.6.3570125.1 $C_6$ (as 6T1) $0$ $1$
1.65.6t1.b.b$1$ $ 5 \cdot 13 $ 6.6.3570125.1 $C_6$ (as 6T1) $0$ $1$
1.13.3t1.a.a$1$ $ 13 $ 3.3.169.1 $C_3$ (as 3T1) $0$ $1$
1.13.3t1.a.b$1$ $ 13 $ 3.3.169.1 $C_3$ (as 3T1) $0$ $1$
* 1.5.4t1.a.a$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
* 1.5.4t1.a.b$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
1.65.12t1.a.a$1$ $ 5 \cdot 13 $ 12.0.1593224064453125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.65.12t1.a.b$1$ $ 5 \cdot 13 $ 12.0.1593224064453125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.65.12t1.a.c$1$ $ 5 \cdot 13 $ 12.0.1593224064453125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.65.12t1.a.d$1$ $ 5 \cdot 13 $ 12.0.1593224064453125.1 $C_{12}$ (as 12T1) $0$ $-1$
2.4225.48.a.a$2$ $ 5^{2} \cdot 13^{2}$ 32.0.24788700386255228556692600250244140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
2.4225.48.a.b$2$ $ 5^{2} \cdot 13^{2}$ 32.0.24788700386255228556692600250244140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
2.4225.48.a.c$2$ $ 5^{2} \cdot 13^{2}$ 32.0.24788700386255228556692600250244140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
2.4225.48.a.d$2$ $ 5^{2} \cdot 13^{2}$ 32.0.24788700386255228556692600250244140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.325.32t402.a.a$2$ $ 5^{2} \cdot 13 $ 32.0.24788700386255228556692600250244140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.325.32t402.a.b$2$ $ 5^{2} \cdot 13 $ 32.0.24788700386255228556692600250244140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.325.32t402.a.c$2$ $ 5^{2} \cdot 13 $ 32.0.24788700386255228556692600250244140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.325.32t402.a.d$2$ $ 5^{2} \cdot 13 $ 32.0.24788700386255228556692600250244140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.325.32t402.a.e$2$ $ 5^{2} \cdot 13 $ 32.0.24788700386255228556692600250244140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.325.32t402.a.f$2$ $ 5^{2} \cdot 13 $ 32.0.24788700386255228556692600250244140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.325.32t402.a.g$2$ $ 5^{2} \cdot 13 $ 32.0.24788700386255228556692600250244140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.325.32t402.a.h$2$ $ 5^{2} \cdot 13 $ 32.0.24788700386255228556692600250244140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 3.4225.4t4.a.a$3$ $ 5^{2} \cdot 13^{2}$ 4.0.4225.1 $A_4$ (as 4T4) $1$ $-1$
* 3.845.6t6.a.a$3$ $ 5 \cdot 13^{2}$ 6.2.142805.1 $A_4\times C_2$ (as 6T6) $1$ $-1$
* 3.21125.12t29.a.a$3$ $ 5^{3} \cdot 13^{2}$ 12.8.1593224064453125.1 $C_4\times A_4$ (as 12T29) $0$ $1$
* 3.21125.12t29.a.b$3$ $ 5^{3} \cdot 13^{2}$ 12.8.1593224064453125.1 $C_4\times A_4$ (as 12T29) $0$ $1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.