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Properties

Label 32.0.48679729995...5376.9
Degree $32$
Signature $[0, 16]$
Discriminant $2^{128}\cdot 3^{16}\cdot 7^{16}$
Root discriminant $73.32$
Ramified primes $2, 3, 7$
Class number $21024$ (GRH)
Class group $[2, 6, 1752]$ (GRH)
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![46058146, -29392688, -66006472, 656503280, -538044580, -1972643120, 3097052168, 200955248, -2353432250, 616194800, 915761288, -811933616, 521407068, -461181840, 412923000, -331507312, 243967235, -165701168, 103821240, -59944080, 31841628, -15521584, 6920264, -2809040, 1032070, -340496, 99848, -25648, 5660, -1040, 152, -16, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 16*x^31 + 152*x^30 - 1040*x^29 + 5660*x^28 - 25648*x^27 + 99848*x^26 - 340496*x^25 + 1032070*x^24 - 2809040*x^23 + 6920264*x^22 - 15521584*x^21 + 31841628*x^20 - 59944080*x^19 + 103821240*x^18 - 165701168*x^17 + 243967235*x^16 - 331507312*x^15 + 412923000*x^14 - 461181840*x^13 + 521407068*x^12 - 811933616*x^11 + 915761288*x^10 + 616194800*x^9 - 2353432250*x^8 + 200955248*x^7 + 3097052168*x^6 - 1972643120*x^5 - 538044580*x^4 + 656503280*x^3 - 66006472*x^2 - 29392688*x + 46058146)
 
gp: K = bnfinit(x^32 - 16*x^31 + 152*x^30 - 1040*x^29 + 5660*x^28 - 25648*x^27 + 99848*x^26 - 340496*x^25 + 1032070*x^24 - 2809040*x^23 + 6920264*x^22 - 15521584*x^21 + 31841628*x^20 - 59944080*x^19 + 103821240*x^18 - 165701168*x^17 + 243967235*x^16 - 331507312*x^15 + 412923000*x^14 - 461181840*x^13 + 521407068*x^12 - 811933616*x^11 + 915761288*x^10 + 616194800*x^9 - 2353432250*x^8 + 200955248*x^7 + 3097052168*x^6 - 1972643120*x^5 - 538044580*x^4 + 656503280*x^3 - 66006472*x^2 - 29392688*x + 46058146, 1)
 

Normalized defining polynomial

\( x^{32} - 16 x^{31} + 152 x^{30} - 1040 x^{29} + 5660 x^{28} - 25648 x^{27} + 99848 x^{26} - 340496 x^{25} + 1032070 x^{24} - 2809040 x^{23} + 6920264 x^{22} - 15521584 x^{21} + 31841628 x^{20} - 59944080 x^{19} + 103821240 x^{18} - 165701168 x^{17} + 243967235 x^{16} - 331507312 x^{15} + 412923000 x^{14} - 461181840 x^{13} + 521407068 x^{12} - 811933616 x^{11} + 915761288 x^{10} + 616194800 x^{9} - 2353432250 x^{8} + 200955248 x^{7} + 3097052168 x^{6} - 1972643120 x^{5} - 538044580 x^{4} + 656503280 x^{3} - 66006472 x^{2} - 29392688 x + 46058146 \)

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:  \(486797299958529611528622959063550881784243829894751113445376=2^{128}\cdot 3^{16}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.32$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(672=2^{5}\cdot 3\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{672}(1,·)$, $\chi_{672}(391,·)$, $\chi_{672}(265,·)$, $\chi_{672}(659,·)$, $\chi_{672}(533,·)$, $\chi_{672}(155,·)$, $\chi_{672}(29,·)$, $\chi_{672}(419,·)$, $\chi_{672}(293,·)$, $\chi_{672}(295,·)$, $\chi_{672}(169,·)$, $\chi_{672}(559,·)$, $\chi_{672}(433,·)$, $\chi_{672}(55,·)$, $\chi_{672}(323,·)$, $\chi_{672}(197,·)$, $\chi_{672}(587,·)$, $\chi_{672}(461,·)$, $\chi_{672}(463,·)$, $\chi_{672}(337,·)$, $\chi_{672}(83,·)$, $\chi_{672}(601,·)$, $\chi_{672}(223,·)$, $\chi_{672}(97,·)$, $\chi_{672}(491,·)$, $\chi_{672}(365,·)$, $\chi_{672}(629,·)$, $\chi_{672}(631,·)$, $\chi_{672}(505,·)$, $\chi_{672}(251,·)$, $\chi_{672}(125,·)$, $\chi_{672}(127,·)$$\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}{2029995739801110497294064068056443278710130635865834523} a^{30} - \frac{5}{676665246600370165764688022685481092903376878621944841} a^{29} - \frac{414516777451659042202746285114911806713003543193586}{2029995739801110497294064068056443278710130635865834523} a^{28} + \frac{233973595431291070141427228765299508769071179839245}{2029995739801110497294064068056443278710130635865834523} a^{27} + \frac{400414227146053235431918256138190673750918487499527}{2029995739801110497294064068056443278710130635865834523} a^{26} + \frac{599574225492554658551142457385630226492820128950350}{2029995739801110497294064068056443278710130635865834523} a^{25} + \frac{1357433412644356677255306125700386266466380293420350}{2029995739801110497294064068056443278710130635865834523} a^{24} - \frac{4070800810217640873735220518324852240951562985257283}{2029995739801110497294064068056443278710130635865834523} a^{23} + \frac{1365470079099744256130174232501193736955288516873689}{676665246600370165764688022685481092903376878621944841} a^{22} - \frac{869660303859313314414103712472782079094041164656822}{2029995739801110497294064068056443278710130635865834523} a^{21} - \frac{1237787880572813446252958244685364687749789592095829}{2029995739801110497294064068056443278710130635865834523} a^{20} + \frac{2373645500677357345049326152782917931392401147598312}{676665246600370165764688022685481092903376878621944841} a^{19} + \frac{2031187873054583714187630041268357148416204403862710}{2029995739801110497294064068056443278710130635865834523} a^{18} - \frac{658913773085007865744890239169478421065869317581871}{2029995739801110497294064068056443278710130635865834523} a^{17} - \frac{1413569370773075690910389119303248425324328175835458}{2029995739801110497294064068056443278710130635865834523} a^{16} - \frac{27546010659238863028532151473998378833799414027923572}{2029995739801110497294064068056443278710130635865834523} a^{15} + \frac{5244374955798573950948600301130580087689269145060022}{2029995739801110497294064068056443278710130635865834523} a^{14} + \frac{2140355576404130735604001225830654370368482011652717}{676665246600370165764688022685481092903376878621944841} a^{13} + \frac{34985162481429919148874882362724835976159252793185006}{2029995739801110497294064068056443278710130635865834523} a^{12} + \frac{34110923214316966791495068551238125957241429219729999}{2029995739801110497294064068056443278710130635865834523} a^{11} - \frac{44249185273311906754493497207991242210352146316351594}{2029995739801110497294064068056443278710130635865834523} a^{10} - \frac{112668512245933032296560540590164763464201145417069785}{2029995739801110497294064068056443278710130635865834523} a^{9} + \frac{3649933048771358249921336108153713650534238613738920}{2029995739801110497294064068056443278710130635865834523} a^{8} + \frac{139493555000799997536111759447135785893227453708876527}{2029995739801110497294064068056443278710130635865834523} a^{7} - \frac{27878553200806751905583032195973697861300804119604794}{2029995739801110497294064068056443278710130635865834523} a^{6} - \frac{124427658081845204406670651301737194136666442497047250}{2029995739801110497294064068056443278710130635865834523} a^{5} + \frac{66796080314832225031682943820992918083991380610357623}{676665246600370165764688022685481092903376878621944841} a^{4} - \frac{109245976086883037863065403085064846413201295917749516}{676665246600370165764688022685481092903376878621944841} a^{3} - \frac{491365574819688707737811867558003014714219112016191104}{2029995739801110497294064068056443278710130635865834523} a^{2} + \frac{145202310964745768661299691190449068702022095905762535}{2029995739801110497294064068056443278710130635865834523} a - \frac{84964054504886353971686868644173236250611680622994772}{225555082200123388588229340895160364301125626207314947}$, $\frac{1}{884563532543257375905816396549164756977134102990919955378915931} a^{31} + \frac{2689793}{10920537438805646616121190080853885888606593864085431547887851} a^{30} + \frac{58219777097673286151860439320080646013818368291063488495003}{884563532543257375905816396549164756977134102990919955378915931} a^{29} + \frac{76661989186078641347911782660082591365863198787993859864193}{884563532543257375905816396549164756977134102990919955378915931} a^{28} + \frac{118485467124840117382305451855692931618456829668929065161292}{294854510847752458635272132183054918992378034330306651792971977} a^{27} - \frac{42928265231168055505794081383529041759753067094241949116021}{98284836949250819545090710727684972997459344776768883930990659} a^{26} + \frac{394481704568091930099781477293714224251502225752171462210231}{884563532543257375905816396549164756977134102990919955378915931} a^{25} - \frac{368143920129112604569897406421880978856010473750616948901231}{884563532543257375905816396549164756977134102990919955378915931} a^{24} + \frac{1813275207369292988091648405302515243523351707820232490114508}{884563532543257375905816396549164756977134102990919955378915931} a^{23} - \frac{242731808816300251840902429149925079112429003097217757591899}{294854510847752458635272132183054918992378034330306651792971977} a^{22} - \frac{682983572529021222464054722991529522737827196552193951312794}{884563532543257375905816396549164756977134102990919955378915931} a^{21} + \frac{1105421839060914563070595751219898717839797604376802642903850}{884563532543257375905816396549164756977134102990919955378915931} a^{20} - \frac{779330035817153890240634358799904716082294990345716800051547}{884563532543257375905816396549164756977134102990919955378915931} a^{19} + \frac{1856428755671618298543561398474629377729673255987157473752004}{884563532543257375905816396549164756977134102990919955378915931} a^{18} + \frac{150038733793815450693223877336574259189474093155486027418873}{98284836949250819545090710727684972997459344776768883930990659} a^{17} + \frac{2460274533705197396292219865488629981668819894945840967726540}{884563532543257375905816396549164756977134102990919955378915931} a^{16} - \frac{11923554408078083803305375430157662146320190267222016569175760}{884563532543257375905816396549164756977134102990919955378915931} a^{15} + \frac{6552592068558445784071601801669922278686862724893761520110612}{884563532543257375905816396549164756977134102990919955378915931} a^{14} - \frac{10648005573919668660375613618497933139879777543698561445210169}{884563532543257375905816396549164756977134102990919955378915931} a^{13} - \frac{14854368177789527674257408750220623641964302512800752955025090}{884563532543257375905816396549164756977134102990919955378915931} a^{12} + \frac{709146695970367376223317853333372675352557776533567933434413}{32761612316416939848363570242561657665819781592256294643663553} a^{11} - \frac{2159160253432806009944103278670151786284900517690092955466434}{294854510847752458635272132183054918992378034330306651792971977} a^{10} - \frac{21643236026534161959737172511267247979687530338375549168638103}{884563532543257375905816396549164756977134102990919955378915931} a^{9} + \frac{22580928482420800657425353517843980774200593552757298387516}{2321689061793326445947024662858700149546283734884304344826551} a^{8} - \frac{102789080952894129097413366326018318958177504264432799644170938}{884563532543257375905816396549164756977134102990919955378915931} a^{7} + \frac{67758709318367620836608399650886535260498669412834826116367816}{884563532543257375905816396549164756977134102990919955378915931} a^{6} + \frac{110608592591395055029164879866090946905712709078787973114929920}{884563532543257375905816396549164756977134102990919955378915931} a^{5} - \frac{92529666246711683156567591409563839404158776736895057543048040}{884563532543257375905816396549164756977134102990919955378915931} a^{4} - \frac{286328877592715749034296503157870011604288894392674695762582366}{884563532543257375905816396549164756977134102990919955378915931} a^{3} + \frac{206953629060550880804164268295023912128690786469707820780776271}{884563532543257375905816396549164756977134102990919955378915931} a^{2} + \frac{430877872295585483316078978940955221854095264525630296195437680}{884563532543257375905816396549164756977134102990919955378915931} a + \frac{432402505776433502212064309790572516510148568677634379131586096}{884563532543257375905816396549164756977134102990919955378915931}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{1752}$, which has order $21024$ (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{8308747870416475249348652429442687054615240}{928210214815322586782836793807244297535496404145329} a^{30} - \frac{124631218056247128740229786441640305819228600}{928210214815322586782836793807244297535496404145329} a^{29} + \frac{3431111842368871127699973835109667259633612748}{2784630644445967760348510381421732892606489212435987} a^{28} - \frac{22735428527746028653532987043882359553567172672}{2784630644445967760348510381421732892606489212435987} a^{27} + \frac{40193551242131525768778932416062647914765453656}{928210214815322586782836793807244297535496404145329} a^{26} - \frac{177845845191793128073872795978955041714474313044}{928210214815322586782836793807244297535496404145329} a^{25} + \frac{677967503514908915632565144072626307646018913746}{928210214815322586782836793807244297535496404145329} a^{24} - \frac{2267147269283571145284075518825871860158634517792}{928210214815322586782836793807244297535496404145329} a^{23} + \frac{6750641341047409387410290901134847556602214436664}{928210214815322586782836793807244297535496404145329} a^{22} - \frac{18069834151673748927882643930197090500257395482808}{928210214815322586782836793807244297535496404145329} a^{21} + \frac{43840243169313968928234770586435982594124410320564}{928210214815322586782836793807244297535496404145329} a^{20} - \frac{96935548625205898960356276783399806489567470004992}{928210214815322586782836793807244297535496404145329} a^{19} + \frac{196281877127457247653293664808914226400015690371784}{928210214815322586782836793807244297535496404145329} a^{18} - \frac{365107092153422730045158403825250282463761891963218}{928210214815322586782836793807244297535496404145329} a^{17} + \frac{625630004173101852679156855569539901774759869566689}{928210214815322586782836793807244297535496404145329} a^{16} - \frac{989066894464138669911160602408066027424554190266912}{928210214815322586782836793807244297535496404145329} a^{15} + \frac{1444690265028979167960480177758041412565566125803432}{928210214815322586782836793807244297535496404145329} a^{14} - \frac{1950346846496011696330418444926440447111673739198904}{928210214815322586782836793807244297535496404145329} a^{13} + \frac{2411377630880418148322486851765014716007437584081284}{928210214815322586782836793807244297535496404145329} a^{12} - \frac{2667895399808080190822937793371706262137873262497344}{928210214815322586782836793807244297535496404145329} a^{11} + \frac{3190093404413643578198957502278562303799442667056344}{928210214815322586782836793807244297535496404145329} a^{10} - \frac{5215906705874830596503563827342577543029952798700756}{928210214815322586782836793807244297535496404145329} a^{9} + \frac{4346023909105821829618049331292028295896289533703570}{928210214815322586782836793807244297535496404145329} a^{8} + \frac{6254153717396227435554391384818405464243945129295648}{928210214815322586782836793807244297535496404145329} a^{7} - \frac{10686143113790639532152861897906620257006061693840776}{928210214815322586782836793807244297535496404145329} a^{6} - \frac{4785147879158732496256017805788749848530650151976632}{928210214815322586782836793807244297535496404145329} a^{5} + \frac{13480208635905943351679933012190481050296459969721780}{928210214815322586782836793807244297535496404145329} a^{4} - \frac{4663675730096572919908148332876867319444312090904192}{928210214815322586782836793807244297535496404145329} a^{3} - \frac{1392642708127843401494564021305249442511941448467063}{2784630644445967760348510381421732892606489212435987} a^{2} - \frac{284422994176842222851743298749241285351906879306306}{2784630644445967760348510381421732892606489212435987} a - \frac{3242292628379579995560059178973124337370087096153177}{2784630644445967760348510381421732892606489212435987} \) (order $16$)
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:  \( 56808797853492.48 \) (assuming GRH)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{7}) \), \(\Q(i, \sqrt{14})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{-2}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{7})\), \(\Q(\sqrt{-2}, \sqrt{-7})\), \(\Q(\zeta_{16})^+\), 4.0.2048.2, 4.0.100352.5, 4.4.100352.1, 8.0.157351936.1, \(\Q(\zeta_{16})\), 8.0.40282095616.2, 8.0.10070523904.2, 8.0.10070523904.1, 8.8.40282095616.1, 8.0.40282095616.1, 8.8.417644767346688.4, 8.0.417644767346688.52, 8.0.173946175488.1, 8.8.173946175488.1, 16.0.1622647227216566419456.1, 16.0.697708606768276588334338277376.8, 16.0.121029087867608368152576.2, 16.0.174427151692069147083584569344.6, 16.0.174427151692069147083584569344.1, 16.16.697708606768276588334338277376.3, 16.0.697708606768276588334338277376.7

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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}$
$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}$