↑

Properties

Label 32.0.15729269092...000.13
Degree $32$
Signature $[0, 16]$
Discriminant $2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}$
Root discriminant $61.29$
Ramified primes $2, 3, 5, 7$
Class number $204800$ (GRH)
Class group $[2, 4, 8, 40, 80]$ (GRH)
Galois group $C_2^3\times C_4$ (as 32T34)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65536, 0, 4718592, 0, 152797184, 0, 511819776, 0, 864389632, 0, 900463872, 0, 640424848, 0, 328468416, 0, 125568321, 0, 36449184, 0, 8103128, 0, 1378608, 0, 177562, 0, 16884, 0, 1129, 0, 48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 48*x^30 + 1129*x^28 + 16884*x^26 + 177562*x^24 + 1378608*x^22 + 8103128*x^20 + 36449184*x^18 + 125568321*x^16 + 328468416*x^14 + 640424848*x^12 + 900463872*x^10 + 864389632*x^8 + 511819776*x^6 + 152797184*x^4 + 4718592*x^2 + 65536)
 
gp: K = bnfinit(x^32 + 48*x^30 + 1129*x^28 + 16884*x^26 + 177562*x^24 + 1378608*x^22 + 8103128*x^20 + 36449184*x^18 + 125568321*x^16 + 328468416*x^14 + 640424848*x^12 + 900463872*x^10 + 864389632*x^8 + 511819776*x^6 + 152797184*x^4 + 4718592*x^2 + 65536, 1)
 

Normalized defining polynomial

\( x^{32} + 48 x^{30} + 1129 x^{28} + 16884 x^{26} + 177562 x^{24} + 1378608 x^{22} + 8103128 x^{20} + 36449184 x^{18} + 125568321 x^{16} + 328468416 x^{14} + 640424848 x^{12} + 900463872 x^{10} + 864389632 x^{8} + 511819776 x^{6} + 152797184 x^{4} + 4718592 x^{2} + 65536 \)

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:  \(1572926909253345615680132503044096000000000000000000000000=2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.29$
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:  \(840=2^{3}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(643,·)$, $\chi_{840}(769,·)$, $\chi_{840}(659,·)$, $\chi_{840}(533,·)$, $\chi_{840}(797,·)$, $\chi_{840}(671,·)$, $\chi_{840}(419,·)$, $\chi_{840}(293,·)$, $\chi_{840}(169,·)$, $\chi_{840}(43,·)$, $\chi_{840}(307,·)$, $\chi_{840}(181,·)$, $\chi_{840}(197,·)$, $\chi_{840}(71,·)$, $\chi_{840}(713,·)$, $\chi_{840}(589,·)$, $\chi_{840}(463,·)$, $\chi_{840}(547,·)$, $\chi_{840}(727,·)$, $\chi_{840}(601,·)$, $\chi_{840}(349,·)$, $\chi_{840}(223,·)$, $\chi_{840}(421,·)$, $\chi_{840}(491,·)$, $\chi_{840}(239,·)$, $\chi_{840}(113,·)$, $\chi_{840}(839,·)$, $\chi_{840}(617,·)$, $\chi_{840}(377,·)$, $\chi_{840}(251,·)$, $\chi_{840}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{4} a^{18} + \frac{1}{4} a^{14} - \frac{1}{2} a^{10} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{19} + \frac{1}{8} a^{15} - \frac{1}{2} a^{13} + \frac{1}{4} a^{11} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{20} - \frac{7}{16} a^{16} + \frac{1}{4} a^{14} - \frac{3}{8} a^{12} - \frac{1}{2} a^{8} + \frac{1}{16} a^{4}$, $\frac{1}{32} a^{21} - \frac{7}{32} a^{17} + \frac{1}{8} a^{15} + \frac{5}{16} a^{13} - \frac{1}{2} a^{11} - \frac{1}{4} a^{9} + \frac{1}{32} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{64} a^{22} - \frac{7}{64} a^{18} + \frac{1}{16} a^{16} - \frac{11}{32} a^{14} + \frac{1}{4} a^{12} - \frac{1}{8} a^{10} - \frac{1}{2} a^{8} + \frac{1}{64} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{128} a^{23} - \frac{7}{128} a^{19} + \frac{1}{32} a^{17} - \frac{11}{64} a^{15} - \frac{3}{8} a^{13} - \frac{1}{16} a^{11} + \frac{1}{4} a^{9} - \frac{63}{128} a^{7} + \frac{1}{8} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{1280} a^{24} + \frac{5}{256} a^{20} + \frac{33}{320} a^{18} + \frac{1}{128} a^{16} + \frac{1}{16} a^{14} + \frac{23}{160} a^{12} + \frac{1}{8} a^{10} - \frac{115}{256} a^{8} - \frac{7}{80} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2} + \frac{1}{5}$, $\frac{1}{2560} a^{25} + \frac{5}{512} a^{21} + \frac{33}{640} a^{19} + \frac{1}{256} a^{17} - \frac{15}{32} a^{15} + \frac{23}{320} a^{13} + \frac{1}{16} a^{11} - \frac{115}{512} a^{9} - \frac{7}{160} a^{7} - \frac{3}{8} a^{5} + \frac{1}{8} a^{3} + \frac{1}{10} a$, $\frac{1}{5120} a^{26} + \frac{5}{1024} a^{22} + \frac{33}{1280} a^{20} + \frac{1}{512} a^{18} - \frac{15}{64} a^{16} - \frac{297}{640} a^{14} + \frac{1}{32} a^{12} + \frac{397}{1024} a^{10} - \frac{7}{320} a^{8} - \frac{3}{16} a^{6} - \frac{7}{16} a^{4} + \frac{1}{20} a^{2}$, $\frac{1}{10240} a^{27} + \frac{5}{2048} a^{23} + \frac{33}{2560} a^{21} + \frac{1}{1024} a^{19} - \frac{15}{128} a^{17} - \frac{297}{1280} a^{15} + \frac{1}{64} a^{13} + \frac{397}{2048} a^{11} - \frac{7}{640} a^{9} + \frac{13}{32} a^{7} - \frac{7}{32} a^{5} - \frac{19}{40} a^{3}$, $\frac{1}{20480} a^{28} - \frac{7}{20480} a^{24} + \frac{33}{5120} a^{22} + \frac{49}{2048} a^{20} - \frac{19}{1280} a^{18} - \frac{177}{2560} a^{16} - \frac{15}{128} a^{14} + \frac{8897}{20480} a^{12} + \frac{313}{1280} a^{10} - \frac{51}{128} a^{8} + \frac{21}{320} a^{6} + \frac{13}{40} a^{4} + \frac{1}{4} a^{2} - \frac{2}{5}$, $\frac{1}{40960} a^{29} - \frac{7}{40960} a^{25} + \frac{33}{10240} a^{23} + \frac{49}{4096} a^{21} - \frac{19}{2560} a^{19} - \frac{177}{5120} a^{17} - \frac{15}{256} a^{15} + \frac{8897}{40960} a^{13} - \frac{967}{2560} a^{11} + \frac{77}{256} a^{9} - \frac{299}{640} a^{7} + \frac{13}{80} a^{5} - \frac{3}{8} a^{3} - \frac{1}{5} a$, $\frac{1}{51629161330411704783754865803575496491745280} a^{30} + \frac{627117687529028444843746790386904481}{42458191883562257223482619904256164878080} a^{28} - \frac{3153091469627575403701437981778571639847}{51629161330411704783754865803575496491745280} a^{26} + \frac{2166227080344787258165672024571088033729}{12907290332602926195938716450893874122936320} a^{24} - \frac{120214365348982044765120907814959251233179}{25814580665205852391877432901787748245872640} a^{22} + \frac{41632760586575724874922380460014787039}{8988363741366940247868186943519410949120} a^{20} - \frac{222852634021905771929974463012605144409241}{6453645166301463097969358225446937061468160} a^{18} - \frac{650626523630206089574372090222338897397983}{1613411291575365774492339556361734265367040} a^{16} + \frac{2955798781152421533466344890480681456575937}{51629161330411704783754865803575496491745280} a^{14} - \frac{593856554211873077812525322166677724444571}{3226822583150731548984679112723468530734080} a^{12} + \frac{562986341052157397393125423657117044761}{25209551430865090226442805568152097896360} a^{10} + \frac{51837321389155139081960260310686874812179}{201676411446920721811542444545216783170880} a^{8} - \frac{40897896000810098435166664719734393983467}{201676411446920721811542444545216783170880} a^{6} + \frac{5831810938077588913422617803048673116821}{12604775715432545113221402784076048948180} a^{4} + \frac{3348490934871636726147779412271898333473}{12604775715432545113221402784076048948180} a^{2} + \frac{446523071088263881010320866026042129661}{3151193928858136278305350696019012237045}$, $\frac{1}{103258322660823409567509731607150992983490560} a^{31} + \frac{627117687529028444843746790386904481}{84916383767124514446965239808512329756160} a^{29} - \frac{3153091469627575403701437981778571639847}{103258322660823409567509731607150992983490560} a^{27} + \frac{2166227080344787258165672024571088033729}{25814580665205852391877432901787748245872640} a^{25} - \frac{120214365348982044765120907814959251233179}{51629161330411704783754865803575496491745280} a^{23} + \frac{41632760586575724874922380460014787039}{17976727482733880495736373887038821898240} a^{21} - \frac{222852634021905771929974463012605144409241}{12907290332602926195938716450893874122936320} a^{19} - \frac{650626523630206089574372090222338897397983}{3226822583150731548984679112723468530734080} a^{17} - \frac{48673362549259283250288520913094815035169343}{103258322660823409567509731607150992983490560} a^{15} - \frac{593856554211873077812525322166677724444571}{6453645166301463097969358225446937061468160} a^{13} + \frac{562986341052157397393125423657117044761}{50419102861730180452885611136304195792720} a^{11} + \frac{51837321389155139081960260310686874812179}{403352822893841443623084889090433566341760} a^{9} - \frac{40897896000810098435166664719734393983467}{403352822893841443623084889090433566341760} a^{7} + \frac{5831810938077588913422617803048673116821}{25209551430865090226442805568152097896360} a^{5} + \frac{3348490934871636726147779412271898333473}{25209551430865090226442805568152097896360} a^{3} - \frac{1352335428884936198647514914996485053692}{3151193928858136278305350696019012237045} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{8}\times C_{40}\times C_{80}$, which has order $204800$ (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:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 11853022514.42188 \) (assuming GRH)
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

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-210}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{2}, \sqrt{-35})\), \(\Q(\sqrt{6}, \sqrt{-35})\), \(\Q(\sqrt{3}, \sqrt{-35})\), \(\Q(\sqrt{3}, \sqrt{-70})\), \(\Q(\sqrt{6}, \sqrt{-70})\), \(\Q(\sqrt{2}, \sqrt{-105})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{10}, \sqrt{-14})\), \(\Q(\sqrt{30}, \sqrt{-35})\), \(\Q(\sqrt{15}, \sqrt{-21})\), \(\Q(\sqrt{5}, \sqrt{-14})\), \(\Q(\sqrt{-7}, \sqrt{10})\), \(\Q(\sqrt{15}, \sqrt{-42})\), \(\Q(\sqrt{-21}, \sqrt{30})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{-21})\), \(\Q(\sqrt{2}, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{-42})\), \(\Q(\sqrt{-7}, \sqrt{30})\), \(\Q(\sqrt{-14}, \sqrt{15})\), \(\Q(\sqrt{10}, \sqrt{-21})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{-7})\), \(\Q(\sqrt{6}, \sqrt{-14})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{3}, \sqrt{-14})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{-21})\), \(\Q(\sqrt{-7}, \sqrt{15})\), \(\Q(\sqrt{-14}, \sqrt{30})\), \(\Q(\sqrt{10}, \sqrt{-42})\), 4.0.98000.1, \(\Q(\zeta_{20})^+\), 4.4.8000.1, 4.0.392000.2, 4.4.72000.1, 4.0.3528000.1, 4.0.55125.1, \(\Q(\zeta_{15})^+\), 8.0.7965941760000.58, 8.0.6146560000.2, 8.0.7965941760000.16, 8.0.497871360000.15, 8.0.7965941760000.52, 8.0.31116960000.6, 8.0.7965941760000.7, 8.0.7965941760000.53, 8.0.7965941760000.43, 8.0.7965941760000.27, 8.0.7965941760000.14, 8.0.7965941760000.62, 8.0.7965941760000.47, 8.8.3317760000.1, 8.0.12745506816.6, 8.0.9604000000.2, 8.0.153664000000.4, 8.0.12446784000000.15, 8.0.3038765625.2, 8.0.2458624000000.2, 8.0.2458624000000.4, 8.0.12446784000000.3, 8.0.12446784000000.16, 8.0.2458624000000.6, \(\Q(\zeta_{40})^+\), 8.8.5184000000.1, 8.0.12446784000000.14, 8.0.199148544000000.142, 8.0.199148544000000.160, 8.0.12446784000000.12, 8.0.12446784000000.13, 8.0.199148544000000.168, 8.8.82944000000.2, 8.8.5184000000.2, 8.0.12446784000000.9, 8.0.777924000000.5, \(\Q(\zeta_{60})^+\), 8.8.82944000000.1, 8.0.199148544000000.137, 8.0.777924000000.7, 8.0.777924000000.6, 8.0.199148544000000.224, 8.0.199148544000000.120, 16.0.63456228123711897600000000.18, 16.0.6044831973376000000000000.2, 16.0.154922431942656000000000000.13, 16.0.39660142577319936000000000000.34, 16.0.154922431942656000000000000.17, 16.0.605165749776000000000000.2, 16.0.39660142577319936000000000000.15, 16.0.39660142577319936000000000000.4, 16.0.39660142577319936000000000000.48, 16.0.39660142577319936000000000000.29, 16.0.39660142577319936000000000000.7, 16.0.39660142577319936000000000000.27, 16.0.39660142577319936000000000000.50, 16.0.39660142577319936000000000000.36, \(\Q(\zeta_{120})^+\)

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 ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$