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Properties

Label 32.0.15729269092...000.12
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 $6400$ (GRH)
Class group $[2, 4, 20, 40]$ (GRH)
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![65536, 0, 196608, 0, 323584, 0, 221184, 0, -507648, 0, 450048, 0, 3558128, 0, 8451504, 0, 11268881, 0, 2112876, 0, 222383, 0, 7032, 0, -1983, 0, 216, 0, 79, 0, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 12*x^30 + 79*x^28 + 216*x^26 - 1983*x^24 + 7032*x^22 + 222383*x^20 + 2112876*x^18 + 11268881*x^16 + 8451504*x^14 + 3558128*x^12 + 450048*x^10 - 507648*x^8 + 221184*x^6 + 323584*x^4 + 196608*x^2 + 65536)
 
gp: K = bnfinit(x^32 + 12*x^30 + 79*x^28 + 216*x^26 - 1983*x^24 + 7032*x^22 + 222383*x^20 + 2112876*x^18 + 11268881*x^16 + 8451504*x^14 + 3558128*x^12 + 450048*x^10 - 507648*x^8 + 221184*x^6 + 323584*x^4 + 196608*x^2 + 65536, 1)
 

Normalized defining polynomial

\( x^{32} + 12 x^{30} + 79 x^{28} + 216 x^{26} - 1983 x^{24} + 7032 x^{22} + 222383 x^{20} + 2112876 x^{18} + 11268881 x^{16} + 8451504 x^{14} + 3558128 x^{12} + 450048 x^{10} - 507648 x^{8} + 221184 x^{6} + 323584 x^{4} + 196608 x^{2} + 65536 \)

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$$\chi_{840}(1,·)$, $\chi_{840}(643,·)$, $\chi_{840}(769,·)$, $\chi_{840}(139,·)$, $\chi_{840}(533,·)$, $\chi_{840}(407,·)$, $\chi_{840}(797,·)$, $\chi_{840}(671,·)$, $\chi_{840}(673,·)$, $\chi_{840}(547,·)$, $\chi_{840}(293,·)$, $\chi_{840}(167,·)$, $\chi_{840}(169,·)$, $\chi_{840}(43,·)$, $\chi_{840}(29,·)$, $\chi_{840}(433,·)$, $\chi_{840}(307,·)$, $\chi_{840}(701,·)$, $\chi_{840}(197,·)$, $\chi_{840}(71,·)$, $\chi_{840}(461,·)$, $\chi_{840}(811,·)$, $\chi_{840}(337,·)$, $\chi_{840}(211,·)$, $\chi_{840}(601,·)$, $\chi_{840}(97,·)$, $\chi_{840}(743,·)$, $\chi_{840}(239,·)$, $\chi_{840}(839,·)$, $\chi_{840}(629,·)$, $\chi_{840}(503,·)$, $\chi_{840}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10} - \frac{1}{11}$, $\frac{1}{11} a^{11} - \frac{1}{11} a$, $\frac{1}{11} a^{12} - \frac{1}{11} a^{2}$, $\frac{1}{11} a^{13} - \frac{1}{11} a^{3}$, $\frac{1}{11} a^{14} - \frac{1}{11} a^{4}$, $\frac{1}{11} a^{15} - \frac{1}{11} a^{5}$, $\frac{1}{11} a^{16} - \frac{1}{11} a^{6}$, $\frac{1}{22} a^{17} - \frac{1}{22} a^{13} - \frac{1}{2} a^{9} + \frac{5}{11} a^{7} - \frac{1}{2} a^{5} - \frac{5}{11} a^{3} - \frac{1}{2} a$, $\frac{1}{44} a^{18} - \frac{1}{44} a^{14} + \frac{1}{44} a^{10} - \frac{3}{11} a^{8} + \frac{1}{4} a^{6} + \frac{3}{11} a^{4} - \frac{1}{4} a^{2} - \frac{3}{11}$, $\frac{1}{88} a^{19} - \frac{1}{88} a^{15} - \frac{1}{22} a^{13} + \frac{1}{88} a^{11} - \frac{3}{22} a^{9} - \frac{3}{8} a^{7} - \frac{4}{11} a^{5} - \frac{7}{88} a^{3} - \frac{3}{22} a$, $\frac{1}{36784} a^{20} - \frac{1}{176} a^{16} + \frac{1}{44} a^{14} + \frac{1}{176} a^{12} + \frac{13}{484} a^{10} - \frac{3}{16} a^{8} + \frac{7}{22} a^{6} - \frac{15}{176} a^{4} - \frac{3}{44} a^{2} - \frac{145}{2299}$, $\frac{1}{73568} a^{21} - \frac{1}{352} a^{17} + \frac{1}{88} a^{15} + \frac{1}{352} a^{13} - \frac{31}{968} a^{11} + \frac{13}{32} a^{9} + \frac{7}{44} a^{7} + \frac{161}{352} a^{5} + \frac{41}{88} a^{3} + \frac{32}{2299} a$, $\frac{1}{147136} a^{22} - \frac{1}{704} a^{18} - \frac{7}{176} a^{16} - \frac{31}{704} a^{14} + \frac{57}{1936} a^{12} + \frac{15}{704} a^{10} + \frac{7}{88} a^{8} + \frac{193}{704} a^{6} - \frac{39}{176} a^{4} - \frac{177}{4598} a^{2} + \frac{2}{11}$, $\frac{1}{294272} a^{23} - \frac{1}{1408} a^{19} - \frac{7}{352} a^{17} - \frac{31}{1408} a^{15} - \frac{119}{3872} a^{13} - \frac{49}{1408} a^{11} - \frac{81}{176} a^{9} + \frac{193}{1408} a^{7} + \frac{137}{352} a^{5} + \frac{241}{9196} a^{3} - \frac{4}{11} a$, $\frac{1}{2942720} a^{24} + \frac{3}{588544} a^{20} - \frac{7}{3520} a^{18} - \frac{51}{2816} a^{16} + \frac{117}{7744} a^{14} - \frac{19}{1280} a^{12} - \frac{63}{3872} a^{10} + \frac{813}{2816} a^{8} + \frac{457}{3520} a^{6} - \frac{668}{2299} a^{4} + \frac{1941}{11495}$, $\frac{1}{5885440} a^{25} + \frac{3}{1177088} a^{21} - \frac{7}{7040} a^{19} - \frac{51}{5632} a^{17} + \frac{117}{15488} a^{15} - \frac{19}{2560} a^{13} - \frac{63}{7744} a^{11} + \frac{813}{5632} a^{9} + \frac{457}{7040} a^{7} + \frac{1631}{4598} a^{5} - \frac{4777}{11495} a$, $\frac{1}{10443065881600} a^{26} - \frac{1657}{59335601600} a^{24} + \frac{9827}{2088613176320} a^{22} + \frac{4886437}{2610766470400} a^{20} - \frac{156532639}{49966822400} a^{18} - \frac{1067687823}{27481752320} a^{16} + \frac{1507591631}{49966822400} a^{14} + \frac{2135590191}{68704380800} a^{12} - \frac{4313898929}{109927009280} a^{10} + \frac{3509884397}{12491705600} a^{8} + \frac{64479202193}{326345808800} a^{6} - \frac{1278995651}{2966780080} a^{4} - \frac{10663334789}{40793226100} a^{2} - \frac{5001146093}{10198306525}$, $\frac{1}{20886131763200} a^{27} - \frac{1657}{118671203200} a^{25} + \frac{9827}{4177226352640} a^{23} + \frac{4886437}{5221532940800} a^{21} - \frac{156532639}{99933644800} a^{19} - \frac{1067687823}{54963504640} a^{17} - \frac{3034846769}{99933644800} a^{15} - \frac{4110262609}{137408761600} a^{13} - \frac{4313898929}{219854018560} a^{11} - \frac{8981821203}{24983411200} a^{9} - \frac{261866606607}{652691617600} a^{7} - \frac{1009288371}{5933560160} a^{5} - \frac{6954859689}{81586452200} a^{3} + \frac{2598580216}{10198306525} a$, $\frac{1}{459494898790400} a^{28} + \frac{1}{22974744939520} a^{26} + \frac{30542351}{459494898790400} a^{24} + \frac{62903373}{28718431174400} a^{22} - \frac{479901971}{91898979758080} a^{20} + \frac{1441865489}{188937047200} a^{18} + \frac{416844414421}{24183942041600} a^{16} - \frac{28613959577}{1209197102080} a^{14} + \frac{731290796491}{24183942041600} a^{12} - \frac{34841950259}{3022992755200} a^{10} + \frac{19265380457}{5743686234880} a^{8} + \frac{1191728425289}{3589803896800} a^{6} + \frac{93875409823}{897450974200} a^{4} + \frac{1438832530}{4487254871} a^{2} - \frac{11633892904}{112181371775}$, $\frac{1}{918989797580800} a^{29} + \frac{1}{45949489879040} a^{27} + \frac{30542351}{918989797580800} a^{25} + \frac{62903373}{57436862348800} a^{23} - \frac{479901971}{183797959516160} a^{21} + \frac{1441865489}{377874094400} a^{19} + \frac{416844414421}{48367884083200} a^{17} + \frac{81313049703}{2418394204160} a^{15} - \frac{1467249389109}{48367884083200} a^{13} - \frac{34841950259}{6045985510400} a^{11} + \frac{19265380457}{11487372469760} a^{9} - \frac{2398075471511}{7179607793600} a^{7} + \frac{12288957623}{1794901948400} a^{5} - \frac{1320245040}{4487254871} a^{3} - \frac{5816946452}{112181371775} a$, $\frac{1}{1837979595161600} a^{30} - \frac{3}{167089054105600} a^{26} + \frac{4654699}{41772263526400} a^{24} + \frac{14165639}{33417810821120} a^{22} - \frac{552467221}{459494898790400} a^{20} - \frac{5776254883}{799469158400} a^{18} + \frac{6212727747}{219854018560} a^{16} + \frac{354810396177}{8794160742400} a^{14} - \frac{45511112871}{2198540185600} a^{12} - \frac{488510516647}{22974744939520} a^{10} - \frac{2569080413}{6245852800} a^{8} + \frac{285130300487}{652691617600} a^{6} - \frac{2147062213}{8158645220} a^{4} + \frac{13548158937}{40793226100} a^{2} + \frac{10641400057}{112181371775}$, $\frac{1}{3675959190323200} a^{31} - \frac{3}{334178108211200} a^{27} + \frac{4654699}{83544527052800} a^{25} + \frac{14165639}{66835621642240} a^{23} - \frac{552467221}{918989797580800} a^{21} - \frac{5776254883}{1598938316800} a^{19} + \frac{6212727747}{439708037120} a^{17} + \frac{354810396177}{17588321484800} a^{15} - \frac{45511112871}{4397080371200} a^{13} + \frac{1600102659673}{45949489879040} a^{11} + \frac{3676772387}{12491705600} a^{9} + \frac{285130300487}{1305383235200} a^{7} - \frac{2147062213}{16317290440} a^{5} + \frac{13548158937}{81586452200} a^{3} + \frac{221546766}{112181371775} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{20}\times C_{40}$, which has order $6400$ (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:  \( -\frac{382579119}{18379795951616} a^{30} - \frac{5385536829}{22974744939520} a^{28} - \frac{26927684145}{18379795951616} a^{26} - \frac{38404921427}{11487372469760} a^{24} + \frac{4061195064837}{91898979758080} a^{22} - \frac{37169032869}{208861317632} a^{20} - \frac{21817280560887}{4836788408320} a^{18} - \frac{49090580796159}{1209197102080} a^{16} - \frac{196591709993013}{967357681664} a^{14} - \frac{1022457410109}{75574818880} a^{12} - \frac{679195652877}{358980389680} a^{10} + \frac{36286157979}{17949019484} a^{8} - \frac{16745193747}{22436274355} a^{6} - \frac{3008991578061}{358980389680} a^{4} - \frac{3296066256}{4487254871} a^{2} - \frac{5650399296}{22436274355} \) (order $10$)
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:  \( 28764339921106.887 \) (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

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{70}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{210}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-2}, \sqrt{-35})\), \(\Q(\sqrt{-6}, \sqrt{-35})\), \(\Q(\sqrt{3}, \sqrt{-35})\), \(\Q(\sqrt{-6}, \sqrt{70})\), \(\Q(\sqrt{3}, \sqrt{70})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{-105})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-10}, \sqrt{14})\), \(\Q(\sqrt{-30}, \sqrt{-35})\), \(\Q(\sqrt{15}, \sqrt{-21})\), \(\Q(\sqrt{5}, \sqrt{14})\), \(\Q(\sqrt{-7}, \sqrt{-10})\), \(\Q(\sqrt{-21}, \sqrt{-30})\), \(\Q(\sqrt{15}, \sqrt{42})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-7})\), \(\Q(\sqrt{-2}, \sqrt{15})\), \(\Q(\sqrt{-2}, \sqrt{-21})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{-6}, \sqrt{-7})\), \(\Q(\sqrt{-6}, \sqrt{14})\), \(\Q(\sqrt{-6}, \sqrt{-10})\), \(\Q(\sqrt{5}, \sqrt{42})\), \(\Q(\sqrt{-7}, \sqrt{-30})\), \(\Q(\sqrt{14}, \sqrt{15})\), \(\Q(\sqrt{-10}, \sqrt{-21})\), \(\Q(\sqrt{5}, \sqrt{-21})\), \(\Q(\sqrt{-7}, \sqrt{15})\), \(\Q(\sqrt{14}, \sqrt{-30})\), \(\Q(\sqrt{-10}, \sqrt{42})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{3}, \sqrt{14})\), \(\Q(\sqrt{3}, \sqrt{-10})\), 4.4.72000.1, 4.0.3528000.1, 4.4.882000.1, 4.0.18000.1, \(\Q(\zeta_{5})\), 4.4.6125.1, 4.0.392000.2, 4.4.8000.1, 8.0.7965941760000.35, 8.0.6146560000.1, 8.0.7965941760000.17, 8.0.497871360000.9, 8.0.7965941760000.51, 8.0.31116960000.6, 8.0.7965941760000.6, 8.0.7965941760000.18, 8.0.7965941760000.11, 8.8.7965941760000.2, 8.0.7965941760000.19, 8.0.3317760000.7, 8.0.12745506816.2, 8.0.7965941760000.55, 8.0.7965941760000.26, 8.0.12446784000000.15, 8.0.777924000000.4, 8.0.37515625.1, 8.0.153664000000.4, 8.8.199148544000000.3, 8.0.199148544000000.163, 8.0.153664000000.6, 8.8.153664000000.2, 8.0.82944000000.5, 8.0.199148544000000.169, 8.0.64000000.1, 8.0.153664000000.3, 8.0.5184000000.5, 8.0.12446784000000.6, 8.0.199148544000000.193, 8.0.82944000000.3, 8.8.12446784000000.4, 8.0.12446784000000.20, 8.8.199148544000000.4, 8.0.199148544000000.220, 8.0.199148544000000.120, 8.0.199148544000000.224, 8.0.777924000000.10, 8.0.777924000000.2, 8.8.82944000000.1, 8.0.199148544000000.137, 8.8.777924000000.1, 8.0.324000000.3, 16.0.63456228123711897600000000.24, 16.0.39660142577319936000000000000.18, 16.0.23612624896000000000000.1, 16.0.154922431942656000000000000.8, 16.0.39660142577319936000000000000.68, 16.0.39660142577319936000000000000.15, 16.0.605165749776000000000000.8, 16.0.39660142577319936000000000000.14, 16.0.39660142577319936000000000000.16, 16.16.39660142577319936000000000000.6, 16.0.39660142577319936000000000000.46, 16.0.6879707136000000000000.7, 16.0.39660142577319936000000000000.21, 16.0.39660142577319936000000000000.40, 16.0.39660142577319936000000000000.59

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 ${\href{/LocalNumberField/11.1.0.1}{1} }^{32}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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}$