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Properties

Label 32.0.22351135421...0000.8
Degree $32$
Signature $[0, 16]$
Discriminant $2^{128}\cdot 3^{16}\cdot 5^{16}$
Root discriminant $61.97$
Ramified primes $2, 3, 5$
Class number Not computed
Class group Not computed
Galois group $C_2^2\times C_8$ (as 32T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![57526849, -143187248, 663710864, -1914237040, 3845008700, -5703032384, 6462476552, -5704396048, 3968694340, -2176666960, 945709544, -315275168, 83608980, -14637840, 2343360, -181936, 29237, 76960, -18600, -58800, 21840, 32240, -19240, -10400, 11050, 208, -3472, 1232, 260, -320, 104, -16, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 16*x^31 + 104*x^30 - 320*x^29 + 260*x^28 + 1232*x^27 - 3472*x^26 + 208*x^25 + 11050*x^24 - 10400*x^23 - 19240*x^22 + 32240*x^21 + 21840*x^20 - 58800*x^19 - 18600*x^18 + 76960*x^17 + 29237*x^16 - 181936*x^15 + 2343360*x^14 - 14637840*x^13 + 83608980*x^12 - 315275168*x^11 + 945709544*x^10 - 2176666960*x^9 + 3968694340*x^8 - 5704396048*x^7 + 6462476552*x^6 - 5703032384*x^5 + 3845008700*x^4 - 1914237040*x^3 + 663710864*x^2 - 143187248*x + 57526849)
 
gp: K = bnfinit(x^32 - 16*x^31 + 104*x^30 - 320*x^29 + 260*x^28 + 1232*x^27 - 3472*x^26 + 208*x^25 + 11050*x^24 - 10400*x^23 - 19240*x^22 + 32240*x^21 + 21840*x^20 - 58800*x^19 - 18600*x^18 + 76960*x^17 + 29237*x^16 - 181936*x^15 + 2343360*x^14 - 14637840*x^13 + 83608980*x^12 - 315275168*x^11 + 945709544*x^10 - 2176666960*x^9 + 3968694340*x^8 - 5704396048*x^7 + 6462476552*x^6 - 5703032384*x^5 + 3845008700*x^4 - 1914237040*x^3 + 663710864*x^2 - 143187248*x + 57526849, 1)
 

Normalized defining polynomial

\( x^{32} - 16 x^{31} + 104 x^{30} - 320 x^{29} + 260 x^{28} + 1232 x^{27} - 3472 x^{26} + 208 x^{25} + 11050 x^{24} - 10400 x^{23} - 19240 x^{22} + 32240 x^{21} + 21840 x^{20} - 58800 x^{19} - 18600 x^{18} + 76960 x^{17} + 29237 x^{16} - 181936 x^{15} + 2343360 x^{14} - 14637840 x^{13} + 83608980 x^{12} - 315275168 x^{11} + 945709544 x^{10} - 2176666960 x^{9} + 3968694340 x^{8} - 5704396048 x^{7} + 6462476552 x^{6} - 5703032384 x^{5} + 3845008700 x^{4} - 1914237040 x^{3} + 663710864 x^{2} - 143187248 x + 57526849 \)

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:  \(2235113542185251937084439154754616201052160000000000000000=2^{128}\cdot 3^{16}\cdot 5^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.97$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(480=2^{5}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{480}(1,·)$, $\chi_{480}(131,·)$, $\chi_{480}(389,·)$, $\chi_{480}(391,·)$, $\chi_{480}(11,·)$, $\chi_{480}(269,·)$, $\chi_{480}(271,·)$, $\chi_{480}(149,·)$, $\chi_{480}(151,·)$, $\chi_{480}(409,·)$, $\chi_{480}(29,·)$, $\chi_{480}(31,·)$, $\chi_{480}(289,·)$, $\chi_{480}(419,·)$, $\chi_{480}(169,·)$, $\chi_{480}(299,·)$, $\chi_{480}(49,·)$, $\chi_{480}(179,·)$, $\chi_{480}(439,·)$, $\chi_{480}(59,·)$, $\chi_{480}(319,·)$, $\chi_{480}(199,·)$, $\chi_{480}(461,·)$, $\chi_{480}(79,·)$, $\chi_{480}(341,·)$, $\chi_{480}(221,·)$, $\chi_{480}(101,·)$, $\chi_{480}(361,·)$, $\chi_{480}(241,·)$, $\chi_{480}(371,·)$, $\chi_{480}(121,·)$, $\chi_{480}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{9} a^{4} + \frac{4}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{9} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{4} - \frac{1}{3} a^{2} - \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{2}{9}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{4}{9} a^{3} - \frac{4}{9} a^{2} + \frac{1}{9}$, $\frac{1}{27} a^{12} + \frac{1}{27} a^{9} - \frac{4}{27} a^{6} + \frac{8}{27} a^{3} + \frac{10}{27}$, $\frac{1}{27} a^{13} + \frac{1}{27} a^{10} - \frac{4}{27} a^{7} - \frac{1}{27} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{8}{27} a - \frac{1}{3}$, $\frac{1}{27} a^{14} + \frac{1}{27} a^{11} - \frac{1}{27} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{4}{27} a^{5} + \frac{1}{9} a^{4} + \frac{1}{9} a^{3} - \frac{11}{27} a^{2} + \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{27} a^{15} + \frac{1}{27} a^{9} - \frac{1}{9} a^{7} + \frac{1}{9} a^{4} - \frac{7}{27} a^{3} + \frac{1}{3} a^{2} + \frac{4}{9} a - \frac{4}{27}$, $\frac{1}{81} a^{16} + \frac{1}{81} a^{15} - \frac{1}{81} a^{14} - \frac{1}{81} a^{12} - \frac{1}{81} a^{11} + \frac{4}{81} a^{10} + \frac{1}{27} a^{9} + \frac{4}{81} a^{8} + \frac{4}{27} a^{7} + \frac{7}{81} a^{6} - \frac{8}{81} a^{5} + \frac{11}{81} a^{4} + \frac{13}{27} a^{3} - \frac{10}{81} a^{2} + \frac{35}{81} a + \frac{34}{81}$, $\frac{1}{81} a^{17} + \frac{1}{81} a^{15} + \frac{1}{81} a^{14} - \frac{1}{81} a^{13} - \frac{4}{81} a^{11} - \frac{1}{81} a^{10} + \frac{4}{81} a^{9} - \frac{1}{81} a^{8} - \frac{5}{81} a^{7} + \frac{1}{27} a^{6} - \frac{8}{81} a^{5} + \frac{1}{81} a^{4} - \frac{34}{81} a^{3} + \frac{4}{9} a^{2} - \frac{1}{81} a - \frac{10}{81}$, $\frac{1}{81} a^{18} - \frac{1}{81} a^{9} + \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{9} a^{4} + \frac{1}{27} a^{3} - \frac{1}{3} a^{2} - \frac{4}{9} a - \frac{22}{81}$, $\frac{1}{81} a^{19} - \frac{1}{81} a^{10} + \frac{1}{9} a^{6} - \frac{2}{27} a^{4} + \frac{2}{9} a^{3} + \frac{1}{3} a^{2} - \frac{31}{81} a - \frac{4}{9}$, $\frac{1}{243} a^{20} - \frac{1}{243} a^{19} - \frac{1}{243} a^{18} + \frac{1}{81} a^{15} - \frac{1}{81} a^{14} - \frac{1}{81} a^{13} + \frac{1}{81} a^{12} - \frac{13}{243} a^{11} + \frac{7}{243} a^{10} - \frac{2}{243} a^{9} - \frac{2}{81} a^{8} + \frac{10}{81} a^{7} - \frac{13}{81} a^{6} - \frac{7}{81} a^{5} - \frac{1}{9} a^{4} + \frac{4}{9} a^{3} - \frac{115}{243} a^{2} + \frac{55}{243} a - \frac{77}{243}$, $\frac{1}{243} a^{21} + \frac{1}{243} a^{19} - \frac{1}{243} a^{18} - \frac{1}{81} a^{15} - \frac{1}{81} a^{14} + \frac{2}{243} a^{12} - \frac{1}{81} a^{11} - \frac{10}{243} a^{10} - \frac{8}{243} a^{9} + \frac{4}{81} a^{8} + \frac{4}{27} a^{7} - \frac{1}{27} a^{6} - \frac{8}{81} a^{5} + \frac{10}{81} a^{4} + \frac{2}{243} a^{3} - \frac{10}{81} a^{2} + \frac{104}{243} a + \frac{46}{243}$, $\frac{1}{243} a^{22} + \frac{1}{243} a^{18} - \frac{1}{81} a^{15} - \frac{4}{243} a^{13} - \frac{4}{81} a^{10} + \frac{5}{243} a^{9} - \frac{7}{81} a^{7} + \frac{1}{9} a^{6} - \frac{37}{243} a^{4} - \frac{28}{81} a^{3} + \frac{20}{81} a + \frac{107}{243}$, $\frac{1}{243} a^{23} + \frac{1}{243} a^{19} + \frac{1}{81} a^{15} + \frac{2}{243} a^{14} - \frac{1}{81} a^{12} - \frac{2}{81} a^{11} - \frac{10}{243} a^{10} + \frac{1}{27} a^{9} + \frac{1}{27} a^{8} + \frac{4}{27} a^{7} + \frac{7}{81} a^{6} + \frac{38}{243} a^{5} + \frac{10}{81} a^{4} + \frac{13}{27} a^{3} + \frac{22}{81} a^{2} + \frac{104}{243} a + \frac{34}{81}$, $\frac{1}{729} a^{24} - \frac{1}{729} a^{21} - \frac{1}{243} a^{19} - \frac{4}{729} a^{18} - \frac{10}{729} a^{15} + \frac{1}{81} a^{14} - \frac{8}{729} a^{12} + \frac{1}{81} a^{11} + \frac{10}{243} a^{10} - \frac{20}{729} a^{9} - \frac{4}{81} a^{8} + \frac{2}{27} a^{7} + \frac{65}{729} a^{6} + \frac{8}{81} a^{5} - \frac{1}{81} a^{4} - \frac{332}{729} a^{3} + \frac{10}{81} a^{2} + \frac{85}{243} a - \frac{146}{729}$, $\frac{1}{729} a^{25} - \frac{1}{729} a^{22} + \frac{2}{729} a^{19} - \frac{1}{243} a^{18} - \frac{1}{729} a^{16} + \frac{1}{81} a^{14} + \frac{10}{729} a^{13} + \frac{1}{81} a^{12} + \frac{1}{81} a^{11} - \frac{26}{729} a^{10} + \frac{13}{243} a^{9} - \frac{4}{81} a^{8} - \frac{88}{729} a^{7} + \frac{11}{81} a^{6} + \frac{8}{81} a^{5} + \frac{10}{729} a^{4} + \frac{25}{81} a^{3} + \frac{37}{81} a^{2} + \frac{163}{729} a + \frac{7}{243}$, $\frac{1}{729} a^{26} - \frac{1}{729} a^{23} - \frac{1}{729} a^{20} + \frac{1}{243} a^{18} - \frac{1}{729} a^{17} - \frac{8}{729} a^{14} - \frac{1}{81} a^{13} - \frac{14}{729} a^{11} - \frac{1}{81} a^{10} - \frac{10}{243} a^{9} + \frac{38}{729} a^{8} + \frac{13}{81} a^{7} - \frac{2}{27} a^{6} + \frac{100}{729} a^{5} + \frac{10}{81} a^{4} - \frac{26}{81} a^{3} + \frac{238}{729} a^{2} + \frac{8}{81} a + \frac{77}{243}$, $\frac{1}{729} a^{27} + \frac{1}{729} a^{21} + \frac{1}{243} a^{19} + \frac{1}{729} a^{18} - \frac{1}{81} a^{14} + \frac{11}{729} a^{12} - \frac{1}{81} a^{11} - \frac{10}{243} a^{10} + \frac{13}{243} a^{9} + \frac{4}{81} a^{8} - \frac{2}{27} a^{7} + \frac{10}{243} a^{6} - \frac{8}{81} a^{5} + \frac{1}{81} a^{4} + \frac{128}{729} a^{3} - \frac{10}{81} a^{2} - \frac{85}{243} a - \frac{125}{729}$, $\frac{1}{2187} a^{28} + \frac{1}{2187} a^{27} - \frac{1}{2187} a^{26} + \frac{1}{2187} a^{25} - \frac{2}{2187} a^{23} - \frac{2}{2187} a^{21} - \frac{2}{2187} a^{20} + \frac{2}{729} a^{19} + \frac{4}{2187} a^{18} - \frac{8}{2187} a^{17} - \frac{10}{2187} a^{16} - \frac{2}{243} a^{15} + \frac{2}{2187} a^{14} - \frac{8}{729} a^{13} - \frac{22}{2187} a^{12} - \frac{55}{2187} a^{11} - \frac{98}{2187} a^{10} + \frac{10}{243} a^{9} - \frac{38}{2187} a^{8} - \frac{202}{2187} a^{7} - \frac{80}{729} a^{6} + \frac{2}{2187} a^{5} - \frac{50}{729} a^{4} + \frac{95}{2187} a^{3} - \frac{739}{2187} a^{2} - \frac{1063}{2187} a - \frac{587}{2187}$, $\frac{1}{2187} a^{29} + \frac{1}{2187} a^{27} - \frac{1}{2187} a^{26} - \frac{1}{2187} a^{25} + \frac{1}{2187} a^{24} - \frac{4}{2187} a^{23} - \frac{2}{2187} a^{22} + \frac{2}{2187} a^{20} - \frac{2}{2187} a^{19} + \frac{2}{729} a^{18} + \frac{1}{2187} a^{17} - \frac{8}{2187} a^{16} + \frac{17}{2187} a^{15} + \frac{7}{2187} a^{14} - \frac{25}{2187} a^{13} - \frac{8}{729} a^{12} - \frac{73}{2187} a^{11} - \frac{82}{2187} a^{10} + \frac{10}{2187} a^{9} - \frac{62}{2187} a^{8} + \frac{70}{2187} a^{7} - \frac{364}{2187} a^{6} - \frac{362}{2187} a^{5} - \frac{214}{2187} a^{4} - \frac{275}{729} a^{3} + \frac{287}{729} a^{2} - \frac{280}{2187} a - \frac{1090}{2187}$, $\frac{1}{207071615826748482093634004169074168864558700754107} a^{30} - \frac{5}{69023871942249494031211334723024722954852900251369} a^{29} + \frac{1032564262959890931617001927228082271356731386}{6679729542798338132052709811905618350469635508197} a^{28} + \frac{4443141530898181775983906751519803903045602091}{7669319104694388225690148302558302550539211139041} a^{27} - \frac{84363464662587622236427532963936885843046041399}{207071615826748482093634004169074168864558700754107} a^{26} + \frac{43808962877818482777092560939884568185123131720}{207071615826748482093634004169074168864558700754107} a^{25} + \frac{62505422439918800619998386609095341711115702332}{207071615826748482093634004169074168864558700754107} a^{24} - \frac{80671540506131909020103312948725582584002037460}{207071615826748482093634004169074168864558700754107} a^{23} + \frac{42833415729055268069320733686833249092134193182}{69023871942249494031211334723024722954852900251369} a^{22} - \frac{31750565411712030454005927439368981231728932366}{69023871942249494031211334723024722954852900251369} a^{21} - \frac{10665462291602759611826149502697454028329376710}{6679729542798338132052709811905618350469635508197} a^{20} - \frac{110941178329616603676299751513875616098609734852}{69023871942249494031211334723024722954852900251369} a^{19} - \frac{279357273997331016547186084998293940601335118861}{207071615826748482093634004169074168864558700754107} a^{18} + \frac{684328569755568949847775784369841507928512339072}{207071615826748482093634004169074168864558700754107} a^{17} + \frac{1058876090956817990092669069258220231759469294688}{207071615826748482093634004169074168864558700754107} a^{16} + \frac{1096438914387360558696845821779429111674204272951}{207071615826748482093634004169074168864558700754107} a^{15} - \frac{2307887676988238661663375330651140637116522993909}{207071615826748482093634004169074168864558700754107} a^{14} - \frac{1114109488735244720300086404172743696415739777009}{69023871942249494031211334723024722954852900251369} a^{13} - \frac{1743221129719327327447964135134175212366049238170}{207071615826748482093634004169074168864558700754107} a^{12} - \frac{5417426686310121989275812852612156459131101790858}{207071615826748482093634004169074168864558700754107} a^{11} - \frac{897901858821245446814009977452480816461775138517}{207071615826748482093634004169074168864558700754107} a^{10} - \frac{427544774535811192013192494792037088109410435092}{207071615826748482093634004169074168864558700754107} a^{9} + \frac{750282886628379760925215974956228077465205514245}{207071615826748482093634004169074168864558700754107} a^{8} + \frac{6148184436522379774064283275821465379652444638197}{207071615826748482093634004169074168864558700754107} a^{7} - \frac{21003597659688643013080582900152008867182360035188}{207071615826748482093634004169074168864558700754107} a^{6} + \frac{21896228989128586561844162431056844495717100307442}{207071615826748482093634004169074168864558700754107} a^{5} - \frac{9672336998991702983674436073237771952005807753182}{69023871942249494031211334723024722954852900251369} a^{4} - \frac{4212350989768669458214018545762493720328800776493}{207071615826748482093634004169074168864558700754107} a^{3} + \frac{566979682273082326235707589285181090092151939307}{207071615826748482093634004169074168864558700754107} a^{2} + \frac{86976857173177770022504662680276528395894888800283}{207071615826748482093634004169074168864558700754107} a + \frac{44267785611432418579140888376161746150524224412858}{207071615826748482093634004169074168864558700754107}$, $\frac{1}{89117891916578078121864205698768126475386062684289479872347} a^{31} + \frac{71728715}{29705963972192692707288068566256042158462020894763159957449} a^{30} + \frac{411881382624815730260880568060272090569669894105754437}{89117891916578078121864205698768126475386062684289479872347} a^{29} - \frac{19458741218646004956107286571271234796225516623317795919}{89117891916578078121864205698768126475386062684289479872347} a^{28} + \frac{35277940387312287860978839102379347558376958162391098884}{89117891916578078121864205698768126475386062684289479872347} a^{27} + \frac{6227913525590578233049233514552991345164603918660615981}{89117891916578078121864205698768126475386062684289479872347} a^{26} + \frac{8341255109624569412657563127909813355413980278031787842}{29705963972192692707288068566256042158462020894763159957449} a^{25} + \frac{28104260931571890394779399342564728640833857769115525419}{89117891916578078121864205698768126475386062684289479872347} a^{24} + \frac{40891887623476981049940308899030574773934962607256771562}{89117891916578078121864205698768126475386062684289479872347} a^{23} + \frac{22041790122269173480026082342361056818630995439016680997}{29705963972192692707288068566256042158462020894763159957449} a^{22} - \frac{14952548779325718363393236993627423607300168864135341992}{29705963972192692707288068566256042158462020894763159957449} a^{21} - \frac{1910088388076349446443598464733134964916618122928399080}{89117891916578078121864205698768126475386062684289479872347} a^{20} - \frac{414528935095466089219510968143042975480881442935394804485}{89117891916578078121864205698768126475386062684289479872347} a^{19} - \frac{105586800075957143102196451289706598947521249269073079785}{29705963972192692707288068566256042158462020894763159957449} a^{18} - \frac{456418277388819340604459721471400747810666674670526954755}{89117891916578078121864205698768126475386062684289479872347} a^{17} - \frac{2247761183039048666528218169279158361241397558438206119}{366740295952996206262815661311802989610642233268680987129} a^{16} - \frac{12964236530472783497926014930388911179574704211787061297}{2874770706986389616834329216089294402431808473686757415237} a^{15} + \frac{597230351774790981743017515206378133031648942233617922485}{89117891916578078121864205698768126475386062684289479872347} a^{14} + \frac{13352706281822305445684950101379179423416114494378198324}{89117891916578078121864205698768126475386062684289479872347} a^{13} + \frac{162360137000571962566195568259698631972629375401173039089}{29705963972192692707288068566256042158462020894763159957449} a^{12} + \frac{4829532745888722276305207033619491863749219213483339269029}{89117891916578078121864205698768126475386062684289479872347} a^{11} + \frac{1153431483736670983186667397590240525880744041976272133377}{89117891916578078121864205698768126475386062684289479872347} a^{10} - \frac{1869602379975620719706240976323908796515573796279353467541}{89117891916578078121864205698768126475386062684289479872347} a^{9} - \frac{4440327616334370653933105658066320050404167094591211661470}{89117891916578078121864205698768126475386062684289479872347} a^{8} + \frac{794363807902301140289949178145371761767093563944618201632}{9901987990730897569096022855418680719487340298254386652483} a^{7} + \frac{9845081699148688775290745514684384231304838043359381986345}{89117891916578078121864205698768126475386062684289479872347} a^{6} - \frac{14209292333222763905633463075873166353500104662582771198631}{89117891916578078121864205698768126475386062684289479872347} a^{5} - \frac{11889297941766521367440799187147326563397791295905205774223}{89117891916578078121864205698768126475386062684289479872347} a^{4} - \frac{14109367822990584237644754106361176675900357914569988399002}{29705963972192692707288068566256042158462020894763159957449} a^{3} + \frac{405001653374358831125200859329726602337640108060628281092}{29705963972192692707288068566256042158462020894763159957449} a^{2} - \frac{11847403484318119598966633136173976277742287319600453362724}{29705963972192692707288068566256042158462020894763159957449} a + \frac{10641343142695414614889888830328015885538362386737713758469}{89117891916578078121864205698768126475386062684289479872347}$

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{23553546258530257807391994115382041625000}{7669319104694388225690148302558302550539211139041} a^{30} + \frac{117767731292651289036959970576910208125000}{2556439701564796075230049434186100850179737046347} a^{29} - \frac{211739598949455516783896636109187773437500}{742192171422037570228078867989513150052181723133} a^{28} + \frac{20174437586839059260702517990049176923750000}{23007957314083164677070444907674907651617633417123} a^{27} - \frac{821959880756085118259579960310372446104930}{852146567188265358410016478062033616726579015449} a^{26} - \frac{5840696100574540051881539777475312833720230}{2556439701564796075230049434186100850179737046347} a^{25} + \frac{73654436009809622233845846650479447398682975}{7669319104694388225690148302558302550539211139041} a^{24} - \frac{23197536785858713403659995599810101758720400}{2556439701564796075230049434186100850179737046347} a^{23} - \frac{161167593619022440951914858520602393899959450}{7669319104694388225690148302558302550539211139041} a^{22} + \frac{470951252393496265579173453156886494408156100}{7669319104694388225690148302558302550539211139041} a^{21} - \frac{244133234619596573348350556961567444687400}{82465796824670841136453207554390350005797969237} a^{20} - \frac{1371022874087756032044140299551753267143436200}{7669319104694388225690148302558302550539211139041} a^{19} + \frac{2871002096984078945665918394538379621012551575}{23007957314083164677070444907674907651617633417123} a^{18} + \frac{963938267745604867499054345590165170943049800}{2556439701564796075230049434186100850179737046347} a^{17} - \frac{2882301644582815158534821263356268796613527275}{7669319104694388225690148302558302550539211139041} a^{16} - \frac{5382156027634500491497525298766312572983360000}{7669319104694388225690148302558302550539211139041} a^{15} + \frac{565085357690498338062379994520712413166872100}{852146567188265358410016478062033616726579015449} a^{14} + \frac{11795310326482423495819814983277748263470818200}{7669319104694388225690148302558302550539211139041} a^{13} - \frac{63032326875534611539297994932537520067280206800}{7669319104694388225690148302558302550539211139041} a^{12} + \frac{94595880449917141003866137566931430288196887200}{2556439701564796075230049434186100850179737046347} a^{11} - \frac{5581045172492727457222698438925397856394810777321}{23007957314083164677070444907674907651617633417123} a^{10} + \frac{6703486176792000954271623510973251816359819635610}{7669319104694388225690148302558302550539211139041} a^{9} - \frac{7266363985959492446826626207739292628606787444620}{2556439701564796075230049434186100850179737046347} a^{8} + \frac{50169102459604667163160546406316851841611570005480}{7669319104694388225690148302558302550539211139041} a^{7} - \frac{10892981771588850096803689866784180214801256119830}{852146567188265358410016478062033616726579015449} a^{6} + \frac{47818774741619010703738327288993223887700281535584}{2556439701564796075230049434186100850179737046347} a^{5} - \frac{163798393105611918482929661358924901850088124630630}{7669319104694388225690148302558302550539211139041} a^{4} + \frac{45079320455557498536322830531374651997631082519960}{2556439701564796075230049434186100850179737046347} a^{3} - \frac{25380496440126029312748067765671999130327483911380}{2556439701564796075230049434186100850179737046347} a^{2} + \frac{77511491025432925441771257086139743865276996797410}{23007957314083164677070444907674907651617633417123} a - \frac{14474647465024231511628859889579722766331624919369}{23007957314083164677070444907674907651617633417123} \) (order $16$)
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

$C_2^2\times C_8$ (as 32T37):

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 $C_2^2\times C_8$
Character table for $C_2^2\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(i, \sqrt{5})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\zeta_{16})^+\), 4.0.2048.2, 4.4.51200.1, 4.0.51200.2, 8.0.40960000.1, \(\Q(\zeta_{16})\), 8.0.10485760000.3, 8.8.2621440000.1, 8.0.2621440000.1, 8.0.10485760000.2, 8.0.10485760000.1, 8.8.108716359680000.1, 8.0.108716359680000.13, 8.8.173946175488.1, 8.0.173946175488.1, 16.0.109951162777600000000.1, 16.0.47276987448284518809600000000.4, 16.0.121029087867608368152576.2, 16.16.11819246862071129702400000000.1, 16.0.11819246862071129702400000000.1, 16.0.47276987448284518809600000000.5, 16.0.47276987448284518809600000000.2

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
5Data not computed