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Properties

Label 32.0.515...121.1
Degree $32$
Signature $[0, 16]$
Discriminant $5.152\times 10^{44}$
Root discriminant $24.96$
Ramified primes $7, 13$
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 - 5*x^31 + 22*x^30 - 73*x^29 + 220*x^28 - 519*x^27 + 1255*x^26 - 2247*x^25 + 4413*x^24 - 6238*x^23 + 9744*x^22 - 10830*x^21 + 13240*x^20 - 10628*x^19 + 12053*x^18 - 5501*x^17 + 12591*x^16 - 3062*x^15 + 17256*x^14 + 194*x^13 + 18165*x^12 + 5659*x^11 + 13232*x^10 - 2318*x^9 - 319*x^8 - 13445*x^7 - 6940*x^6 - 4990*x^5 + 1466*x^4 + 2074*x^3 + 1626*x^2 + 264*x + 79)
 
gp: K = bnfinit(x^32 - 5*x^31 + 22*x^30 - 73*x^29 + 220*x^28 - 519*x^27 + 1255*x^26 - 2247*x^25 + 4413*x^24 - 6238*x^23 + 9744*x^22 - 10830*x^21 + 13240*x^20 - 10628*x^19 + 12053*x^18 - 5501*x^17 + 12591*x^16 - 3062*x^15 + 17256*x^14 + 194*x^13 + 18165*x^12 + 5659*x^11 + 13232*x^10 - 2318*x^9 - 319*x^8 - 13445*x^7 - 6940*x^6 - 4990*x^5 + 1466*x^4 + 2074*x^3 + 1626*x^2 + 264*x + 79, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![79, 264, 1626, 2074, 1466, -4990, -6940, -13445, -319, -2318, 13232, 5659, 18165, 194, 17256, -3062, 12591, -5501, 12053, -10628, 13240, -10830, 9744, -6238, 4413, -2247, 1255, -519, 220, -73, 22, -5, 1]);
 

\( x^{32} - 5 x^{31} + 22 x^{30} - 73 x^{29} + 220 x^{28} - 519 x^{27} + 1255 x^{26} - 2247 x^{25} + 4413 x^{24} - 6238 x^{23} + 9744 x^{22} - 10830 x^{21} + 13240 x^{20} - 10628 x^{19} + 12053 x^{18} - 5501 x^{17} + 12591 x^{16} - 3062 x^{15} + 17256 x^{14} + 194 x^{13} + 18165 x^{12} + 5659 x^{11} + 13232 x^{10} - 2318 x^{9} - 319 x^{8} - 13445 x^{7} - 6940 x^{6} - 4990 x^{5} + 1466 x^{4} + 2074 x^{3} + 1626 x^{2} + 264 x + 79 \)

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:  \(515207889456069254383620937908064472169867121\)\(\medspace = 7^{16}\cdot 13^{28}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $24.96$
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:  $7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $\frac{1}{3} a^{29} + \frac{1}{3} a^{28} + \frac{1}{3} a^{27} + \frac{1}{3} a^{26} - \frac{1}{3} a^{24} + \frac{1}{3} a^{21} - \frac{1}{3} a^{20} - \frac{1}{3} a^{18} - \frac{1}{3} a^{17} + \frac{1}{3} a^{16} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{387} a^{30} + \frac{47}{387} a^{29} - \frac{190}{387} a^{28} - \frac{109}{387} a^{27} - \frac{38}{387} a^{26} - \frac{145}{387} a^{25} - \frac{4}{9} a^{24} + \frac{29}{129} a^{23} - \frac{179}{387} a^{22} - \frac{11}{43} a^{21} + \frac{140}{387} a^{20} + \frac{104}{387} a^{19} + \frac{139}{387} a^{18} + \frac{59}{129} a^{17} - \frac{14}{387} a^{16} + \frac{56}{387} a^{15} + \frac{55}{387} a^{14} + \frac{19}{129} a^{13} - \frac{85}{387} a^{12} - \frac{44}{387} a^{11} + \frac{149}{387} a^{10} - \frac{148}{387} a^{9} - \frac{112}{387} a^{8} - \frac{119}{387} a^{7} - \frac{59}{387} a^{6} + \frac{13}{387} a^{5} - \frac{103}{387} a^{4} + \frac{2}{43} a^{3} + \frac{13}{129} a^{2} + \frac{25}{387} a - \frac{38}{387}$, $\frac{1}{18124710121459973816040879249892477768662471870064700049002303796359413719} a^{31} + \frac{98712795513458524901771244955570036915150026230522244430707188477555}{6041570040486657938680293083297492589554157290021566683000767932119804573} a^{30} + \frac{121228754910753155662923612321520759840096709461283798291426283784532953}{18124710121459973816040879249892477768662471870064700049002303796359413719} a^{29} + \frac{3919541331197629290243888022175219006485158841094257702417095758691637846}{18124710121459973816040879249892477768662471870064700049002303796359413719} a^{28} - \frac{1974232948449067371665365032987537195606032411158675474264600068357475349}{6041570040486657938680293083297492589554157290021566683000767932119804573} a^{27} + \frac{442940631966517255221725879630083418085161143341494138700115907624885993}{2013856680162219312893431027765830863184719096673855561000255977373268191} a^{26} - \frac{5101484628867481205082069808564800777007719244515985988106744343188691106}{18124710121459973816040879249892477768662471870064700049002303796359413719} a^{25} - \frac{6169728047816107940686838128138270738544168876077369177763188392387219541}{18124710121459973816040879249892477768662471870064700049002303796359413719} a^{24} + \frac{784857859375439740664520043905181131910953376515554831129704504416580339}{18124710121459973816040879249892477768662471870064700049002303796359413719} a^{23} + \frac{4321345895738681923576778166176426243991057659822865862323188857643657803}{18124710121459973816040879249892477768662471870064700049002303796359413719} a^{22} + \frac{1534686409583999858227404494465021519816443937727528865919217536558661809}{18124710121459973816040879249892477768662471870064700049002303796359413719} a^{21} + \frac{6915304050351639712304384045905623335207800104285307229456933703135403888}{18124710121459973816040879249892477768662471870064700049002303796359413719} a^{20} - \frac{286520331084094536434499287050233730077754949552578535600505801502177420}{6041570040486657938680293083297492589554157290021566683000767932119804573} a^{19} + \frac{961503829173312440222487008484282132694453062368406250099312296857124331}{18124710121459973816040879249892477768662471870064700049002303796359413719} a^{18} - \frac{8678112088143920079258380033979598538752293902362491345140987554269009854}{18124710121459973816040879249892477768662471870064700049002303796359413719} a^{17} - \frac{6328187400239140390778869138475407137723859570395278836686461456040743}{24862428150150855714733716392170751397342211070047599518521678733003311} a^{16} + \frac{811604452652777595925252861253766090708461443621889728140406526527977436}{2013856680162219312893431027765830863184719096673855561000255977373268191} a^{15} + \frac{4959284153814176018913637465884749003242687325365116430881417066478491034}{18124710121459973816040879249892477768662471870064700049002303796359413719} a^{14} + \frac{7953473577258474740509709385847204309042381434306878032753682519826821039}{18124710121459973816040879249892477768662471870064700049002303796359413719} a^{13} - \frac{1183094637895091556663157845896350386924435293478804071855212865938203488}{6041570040486657938680293083297492589554157290021566683000767932119804573} a^{12} - \frac{34111856171557385881440776077082795246152583548952610877337525950338725}{671285560054073104297810342588610287728239698891285187000085325791089397} a^{11} - \frac{5141561395011130140519618147185992304924232323272204359219557697078303956}{18124710121459973816040879249892477768662471870064700049002303796359413719} a^{10} + \frac{207415374130655698880642170874254718910672246421552304716556526485333017}{421504886545580786419555331392848320201452834187551163930286134799056133} a^{9} + \frac{229434842787857674604759611695471891498768032789539249596942364614314561}{2013856680162219312893431027765830863184719096673855561000255977373268191} a^{8} - \frac{7080017637282973974535393916535141641758170089881487729477134277632719517}{18124710121459973816040879249892477768662471870064700049002303796359413719} a^{7} - \frac{7863822310876602459095399436261711688805841901843161751116791974604554699}{18124710121459973816040879249892477768662471870064700049002303796359413719} a^{6} + \frac{2644104050468948175798110555010687477876968202840951088480354588929638326}{6041570040486657938680293083297492589554157290021566683000767932119804573} a^{5} + \frac{856475944101239793114733590461524704025961042025170416947205634543911802}{18124710121459973816040879249892477768662471870064700049002303796359413719} a^{4} - \frac{2000672211116484955071110638878853998874384010465730259548325392533582354}{6041570040486657938680293083297492589554157290021566683000767932119804573} a^{3} + \frac{2786087472603381630559296012172991621692412650935899373047742010971864718}{18124710121459973816040879249892477768662471870064700049002303796359413719} a^{2} + \frac{1661492937590189995847628865878848624678738457914778080329150747567670168}{18124710121459973816040879249892477768662471870064700049002303796359413719} a - \frac{88952712973756918332976590862151443626133556002853324699503798939538555}{229426710398227516658745306960664275552689517342591139860788655650119161}$

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:  \( -1 \) (order $2$)
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:  \( 1483068056.9314811 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 1483068056.9314811 \cdot 1}{2\sqrt{515207889456069254383620937908064472169867121}}\approx 0.192760359061108$ (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

\(\Q(\sqrt{13}) \), 4.0.2197.1, 4.0.8281.1, 8.0.68574961.1, 16.0.134308824412163591281.1

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 $24{,}\,{\href{/LocalNumberField/2.8.0.1}{8} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ $24{,}\,{\href{/LocalNumberField/5.8.0.1}{8} }$ R $24{,}\,{\href{/LocalNumberField/11.8.0.1}{8} }$ R ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ $24{,}\,{\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ $24{,}\,{\href{/LocalNumberField/31.8.0.1}{8} }$ $24{,}\,{\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{16}$ $24{,}\,{\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ $24{,}\,{\href{/LocalNumberField/59.8.0.1}{8} }$

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
7Data not computed
13Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.13.2t1.a.a$1$ $ 13 $ \(\Q(\sqrt{13}) \) $C_2$ (as 2T1) $1$ $1$
1.91.6t1.j.a$1$ $ 7 \cdot 13 $ 6.6.5274997.1 $C_6$ (as 6T1) $0$ $1$
1.91.6t1.j.b$1$ $ 7 \cdot 13 $ 6.6.5274997.1 $C_6$ (as 6T1) $0$ $1$
1.7.3t1.a.a$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
1.7.3t1.a.b$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
* 1.13.4t1.a.a$1$ $ 13 $ 4.0.2197.1 $C_4$ (as 4T1) $0$ $-1$
* 1.13.4t1.a.b$1$ $ 13 $ 4.0.2197.1 $C_4$ (as 4T1) $0$ $-1$
1.91.12t1.a.a$1$ $ 7 \cdot 13 $ 12.0.61132828589969773.1 $C_{12}$ (as 12T1) $0$ $-1$
1.91.12t1.a.b$1$ $ 7 \cdot 13 $ 12.0.61132828589969773.1 $C_{12}$ (as 12T1) $0$ $-1$
1.91.12t1.a.c$1$ $ 7 \cdot 13 $ 12.0.61132828589969773.1 $C_{12}$ (as 12T1) $0$ $-1$
1.91.12t1.a.d$1$ $ 7 \cdot 13 $ 12.0.61132828589969773.1 $C_{12}$ (as 12T1) $0$ $-1$
2.8281.48.a.a$2$ $ 7^{2} \cdot 13^{2}$ 32.0.515207889456069254383620937908064472169867121.1 $C_8.A_4$ (as 32T402) $0$ $0$
2.8281.48.a.b$2$ $ 7^{2} \cdot 13^{2}$ 32.0.515207889456069254383620937908064472169867121.1 $C_8.A_4$ (as 32T402) $0$ $0$
2.8281.48.a.c$2$ $ 7^{2} \cdot 13^{2}$ 32.0.515207889456069254383620937908064472169867121.1 $C_8.A_4$ (as 32T402) $0$ $0$
2.8281.48.a.d$2$ $ 7^{2} \cdot 13^{2}$ 32.0.515207889456069254383620937908064472169867121.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.1183.32t402.a.a$2$ $ 7 \cdot 13^{2}$ 32.0.515207889456069254383620937908064472169867121.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.1183.32t402.a.b$2$ $ 7 \cdot 13^{2}$ 32.0.515207889456069254383620937908064472169867121.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.1183.32t402.a.c$2$ $ 7 \cdot 13^{2}$ 32.0.515207889456069254383620937908064472169867121.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.1183.32t402.a.d$2$ $ 7 \cdot 13^{2}$ 32.0.515207889456069254383620937908064472169867121.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.1183.32t402.a.e$2$ $ 7 \cdot 13^{2}$ 32.0.515207889456069254383620937908064472169867121.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.1183.32t402.a.f$2$ $ 7 \cdot 13^{2}$ 32.0.515207889456069254383620937908064472169867121.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.1183.32t402.a.g$2$ $ 7 \cdot 13^{2}$ 32.0.515207889456069254383620937908064472169867121.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.1183.32t402.a.h$2$ $ 7 \cdot 13^{2}$ 32.0.515207889456069254383620937908064472169867121.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 3.8281.4t4.b.a$3$ $ 7^{2} \cdot 13^{2}$ 4.0.8281.1 $A_4$ (as 4T4) $1$ $-1$
* 3.637.6t6.a.a$3$ $ 7^{2} \cdot 13 $ 6.2.31213.1 $A_4\times C_2$ (as 6T6) $1$ $-1$
* 3.107653.12t29.a.a$3$ $ 7^{2} \cdot 13^{3}$ 12.8.61132828589969773.1 $C_4\times A_4$ (as 12T29) $0$ $1$
* 3.107653.12t29.a.b$3$ $ 7^{2} \cdot 13^{3}$ 12.8.61132828589969773.1 $C_4\times A_4$ (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 *.