Normalized defining polynomial
\( x^{32} + 16 x^{30} + 152 x^{28} + 960 x^{26} + 4525 x^{24} + 16184 x^{22} + 45376 x^{20} + 98960 x^{18} + 169224 x^{16} + 220120 x^{14} + 215964 x^{12} + 146912 x^{10} + \cdots + 1 \)
Invariants
Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(203282392447840896882957090816000000000000000000000000\) \(\medspace = 2^{96}\cdot 3^{16}\cdot 5^{24}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(46.33\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $2^{3}3^{1/2}5^{3/4}\approx 46.331687411530766$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $32$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(240=2^{4}\cdot 3\cdot 5\) | ||
Dirichlet character group: | $\lbrace$$\chi_{240}(1,·)$, $\chi_{240}(7,·)$, $\chi_{240}(143,·)$, $\chi_{240}(149,·)$, $\chi_{240}(23,·)$, $\chi_{240}(29,·)$, $\chi_{240}(161,·)$, $\chi_{240}(163,·)$, $\chi_{240}(167,·)$, $\chi_{240}(41,·)$, $\chi_{240}(43,·)$, $\chi_{240}(47,·)$, $\chi_{240}(49,·)$, $\chi_{240}(181,·)$, $\chi_{240}(187,·)$, $\chi_{240}(61,·)$, $\chi_{240}(67,·)$, $\chi_{240}(203,·)$, $\chi_{240}(209,·)$, $\chi_{240}(83,·)$, $\chi_{240}(89,·)$, $\chi_{240}(221,·)$, $\chi_{240}(223,·)$, $\chi_{240}(227,·)$, $\chi_{240}(101,·)$, $\chi_{240}(103,·)$, $\chi_{240}(107,·)$, $\chi_{240}(109,·)$, $\chi_{240}(229,·)$, $\chi_{240}(169,·)$, $\chi_{240}(121,·)$, $\chi_{240}(127,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{32768}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{4}a^{24}+\frac{1}{4}$, $\frac{1}{4}a^{25}+\frac{1}{4}a$, $\frac{1}{4}a^{26}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{27}+\frac{1}{4}a^{3}$, $\frac{1}{796}a^{28}+\frac{21}{398}a^{26}-\frac{37}{398}a^{24}+\frac{49}{199}a^{22}-\frac{27}{199}a^{20}+\frac{54}{199}a^{18}-\frac{23}{199}a^{16}-\frac{66}{199}a^{14}+\frac{60}{199}a^{12}+\frac{99}{199}a^{10}-\frac{28}{199}a^{8}+\frac{43}{199}a^{6}-\frac{283}{796}a^{4}+\frac{139}{398}a^{2}+\frac{65}{398}$, $\frac{1}{796}a^{29}+\frac{21}{398}a^{27}-\frac{37}{398}a^{25}+\frac{49}{199}a^{23}-\frac{27}{199}a^{21}+\frac{54}{199}a^{19}-\frac{23}{199}a^{17}-\frac{66}{199}a^{15}+\frac{60}{199}a^{13}+\frac{99}{199}a^{11}-\frac{28}{199}a^{9}+\frac{43}{199}a^{7}-\frac{283}{796}a^{5}+\frac{139}{398}a^{3}+\frac{65}{398}a$, $\frac{1}{33\!\cdots\!36}a^{30}+\frac{87\!\cdots\!73}{33\!\cdots\!36}a^{28}+\frac{19\!\cdots\!25}{16\!\cdots\!18}a^{26}+\frac{29\!\cdots\!75}{33\!\cdots\!36}a^{24}-\frac{34\!\cdots\!30}{83\!\cdots\!59}a^{22}-\frac{13\!\cdots\!91}{83\!\cdots\!59}a^{20}-\frac{21\!\cdots\!01}{83\!\cdots\!59}a^{18}+\frac{24\!\cdots\!97}{83\!\cdots\!59}a^{16}+\frac{24\!\cdots\!50}{83\!\cdots\!59}a^{14}+\frac{28\!\cdots\!21}{83\!\cdots\!59}a^{12}-\frac{38\!\cdots\!57}{83\!\cdots\!59}a^{10}+\frac{64\!\cdots\!53}{83\!\cdots\!59}a^{8}+\frac{15\!\cdots\!09}{33\!\cdots\!36}a^{6}+\frac{16\!\cdots\!41}{33\!\cdots\!36}a^{4}-\frac{72\!\cdots\!85}{16\!\cdots\!18}a^{2}+\frac{16\!\cdots\!91}{33\!\cdots\!36}$, $\frac{1}{33\!\cdots\!36}a^{31}+\frac{87\!\cdots\!73}{33\!\cdots\!36}a^{29}+\frac{19\!\cdots\!25}{16\!\cdots\!18}a^{27}+\frac{29\!\cdots\!75}{33\!\cdots\!36}a^{25}-\frac{34\!\cdots\!30}{83\!\cdots\!59}a^{23}-\frac{13\!\cdots\!91}{83\!\cdots\!59}a^{21}-\frac{21\!\cdots\!01}{83\!\cdots\!59}a^{19}+\frac{24\!\cdots\!97}{83\!\cdots\!59}a^{17}+\frac{24\!\cdots\!50}{83\!\cdots\!59}a^{15}+\frac{28\!\cdots\!21}{83\!\cdots\!59}a^{13}-\frac{38\!\cdots\!57}{83\!\cdots\!59}a^{11}+\frac{64\!\cdots\!53}{83\!\cdots\!59}a^{9}+\frac{15\!\cdots\!09}{33\!\cdots\!36}a^{7}+\frac{16\!\cdots\!41}{33\!\cdots\!36}a^{5}-\frac{72\!\cdots\!85}{16\!\cdots\!18}a^{3}+\frac{16\!\cdots\!91}{33\!\cdots\!36}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{4}\times C_{8}\times C_{80}$, which has order $2560$ (assuming GRH)
Unit group
Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{4366251400675137246941}{166798748783521780363118} a^{30} - \frac{139419596465868338323055}{333597497567043560726236} a^{28} - \frac{330649311330808604060600}{83399374391760890181559} a^{26} - \frac{4169267722004324704631585}{166798748783521780363118} a^{24} - \frac{9808880326525928344379500}{83399374391760890181559} a^{22} - \frac{35006374090433504431991038}{83399374391760890181559} a^{20} - \frac{97913368924490331152866680}{83399374391760890181559} a^{18} - \frac{212870332447547083767334180}{83399374391760890181559} a^{16} - \frac{362653308169359271622909280}{83399374391760890181559} a^{14} - \frac{469216208282469529517110860}{83399374391760890181559} a^{12} - \frac{457227468859736880583674912}{83399374391760890181559} a^{10} - \frac{307351339084823514267820220}{83399374391760890181559} a^{8} - \frac{277212667649755983508367585}{166798748783521780363118} a^{6} - \frac{107263433800862459772757275}{333597497567043560726236} a^{4} - \frac{4042882751195651204983510}{83399374391760890181559} a^{2} - \frac{28386804048725376514543}{166798748783521780363118} \) (order $6$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{10\!\cdots\!91}{83\!\cdots\!59}a^{30}+\frac{33\!\cdots\!61}{16\!\cdots\!18}a^{28}+\frac{15\!\cdots\!28}{83\!\cdots\!59}a^{26}+\frac{10\!\cdots\!53}{83\!\cdots\!59}a^{24}+\frac{47\!\cdots\!16}{83\!\cdots\!59}a^{22}+\frac{17\!\cdots\!44}{83\!\cdots\!59}a^{20}+\frac{48\!\cdots\!72}{83\!\cdots\!59}a^{18}+\frac{10\!\cdots\!17}{83\!\cdots\!59}a^{16}+\frac{18\!\cdots\!84}{83\!\cdots\!59}a^{14}+\frac{24\!\cdots\!96}{83\!\cdots\!59}a^{12}+\frac{24\!\cdots\!20}{83\!\cdots\!59}a^{10}+\frac{17\!\cdots\!68}{83\!\cdots\!59}a^{8}+\frac{84\!\cdots\!11}{83\!\cdots\!59}a^{6}+\frac{41\!\cdots\!97}{16\!\cdots\!18}a^{4}+\frac{25\!\cdots\!68}{83\!\cdots\!59}a^{2}+\frac{62\!\cdots\!85}{83\!\cdots\!59}$, $\frac{85\!\cdots\!79}{33\!\cdots\!36}a^{30}+\frac{34\!\cdots\!35}{83\!\cdots\!59}a^{28}+\frac{32\!\cdots\!87}{83\!\cdots\!59}a^{26}+\frac{20\!\cdots\!64}{83\!\cdots\!59}a^{24}+\frac{96\!\cdots\!13}{83\!\cdots\!59}a^{22}+\frac{34\!\cdots\!62}{83\!\cdots\!59}a^{20}+\frac{97\!\cdots\!20}{83\!\cdots\!59}a^{18}+\frac{21\!\cdots\!90}{83\!\cdots\!59}a^{16}+\frac{36\!\cdots\!78}{83\!\cdots\!59}a^{14}+\frac{47\!\cdots\!98}{83\!\cdots\!59}a^{12}+\frac{47\!\cdots\!87}{83\!\cdots\!59}a^{10}+\frac{32\!\cdots\!32}{83\!\cdots\!59}a^{8}+\frac{60\!\cdots\!67}{33\!\cdots\!36}a^{6}+\frac{32\!\cdots\!67}{83\!\cdots\!59}a^{4}+\frac{45\!\cdots\!16}{83\!\cdots\!59}a^{2}+\frac{10\!\cdots\!96}{83\!\cdots\!59}$, $\frac{17\!\cdots\!07}{33\!\cdots\!36}a^{30}+\frac{13\!\cdots\!23}{16\!\cdots\!18}a^{28}+\frac{13\!\cdots\!25}{16\!\cdots\!18}a^{26}+\frac{81\!\cdots\!45}{16\!\cdots\!18}a^{24}+\frac{19\!\cdots\!66}{83\!\cdots\!59}a^{22}+\frac{68\!\cdots\!48}{83\!\cdots\!59}a^{20}+\frac{19\!\cdots\!60}{83\!\cdots\!59}a^{18}+\frac{41\!\cdots\!48}{83\!\cdots\!59}a^{16}+\frac{71\!\cdots\!60}{83\!\cdots\!59}a^{14}+\frac{92\!\cdots\!70}{83\!\cdots\!59}a^{12}+\frac{90\!\cdots\!48}{83\!\cdots\!59}a^{10}+\frac{60\!\cdots\!20}{83\!\cdots\!59}a^{8}+\frac{10\!\cdots\!35}{33\!\cdots\!36}a^{6}+\frac{10\!\cdots\!87}{16\!\cdots\!18}a^{4}+\frac{15\!\cdots\!15}{16\!\cdots\!18}a^{2}+\frac{38\!\cdots\!81}{16\!\cdots\!18}$, $\frac{80\!\cdots\!25}{33\!\cdots\!36}a^{31}+\frac{64\!\cdots\!71}{16\!\cdots\!18}a^{29}+\frac{61\!\cdots\!89}{16\!\cdots\!18}a^{27}+\frac{38\!\cdots\!63}{16\!\cdots\!18}a^{25}+\frac{90\!\cdots\!90}{83\!\cdots\!59}a^{23}+\frac{32\!\cdots\!36}{83\!\cdots\!59}a^{21}+\frac{91\!\cdots\!60}{83\!\cdots\!59}a^{19}+\frac{19\!\cdots\!07}{83\!\cdots\!59}a^{17}+\frac{33\!\cdots\!48}{83\!\cdots\!59}a^{15}+\frac{44\!\cdots\!58}{83\!\cdots\!59}a^{13}+\frac{43\!\cdots\!59}{83\!\cdots\!59}a^{11}+\frac{29\!\cdots\!12}{83\!\cdots\!59}a^{9}+\frac{53\!\cdots\!97}{33\!\cdots\!36}a^{7}+\frac{54\!\cdots\!49}{16\!\cdots\!18}a^{5}+\frac{79\!\cdots\!87}{16\!\cdots\!18}a^{3}+\frac{19\!\cdots\!75}{16\!\cdots\!18}a$, $\frac{25\!\cdots\!89}{33\!\cdots\!36}a^{31}+\frac{41\!\cdots\!01}{33\!\cdots\!36}a^{29}+\frac{19\!\cdots\!25}{16\!\cdots\!18}a^{27}+\frac{61\!\cdots\!65}{83\!\cdots\!59}a^{25}+\frac{29\!\cdots\!66}{83\!\cdots\!59}a^{23}+\frac{10\!\cdots\!86}{83\!\cdots\!59}a^{21}+\frac{29\!\cdots\!40}{83\!\cdots\!59}a^{19}+\frac{63\!\cdots\!28}{83\!\cdots\!59}a^{17}+\frac{10\!\cdots\!40}{83\!\cdots\!59}a^{15}+\frac{13\!\cdots\!30}{83\!\cdots\!59}a^{13}+\frac{13\!\cdots\!60}{83\!\cdots\!59}a^{11}+\frac{91\!\cdots\!40}{83\!\cdots\!59}a^{9}+\frac{16\!\cdots\!05}{33\!\cdots\!36}a^{7}+\frac{32\!\cdots\!49}{33\!\cdots\!36}a^{5}+\frac{24\!\cdots\!35}{16\!\cdots\!18}a^{3}+\frac{29\!\cdots\!21}{83\!\cdots\!59}a$, $\frac{13\!\cdots\!05}{16\!\cdots\!18}a^{31}+\frac{10\!\cdots\!50}{83\!\cdots\!59}a^{29}+\frac{99\!\cdots\!84}{83\!\cdots\!59}a^{27}+\frac{12\!\cdots\!89}{16\!\cdots\!18}a^{25}+\frac{29\!\cdots\!48}{83\!\cdots\!59}a^{23}+\frac{10\!\cdots\!44}{83\!\cdots\!59}a^{21}+\frac{28\!\cdots\!38}{83\!\cdots\!59}a^{19}+\frac{61\!\cdots\!64}{83\!\cdots\!59}a^{17}+\frac{10\!\cdots\!88}{83\!\cdots\!59}a^{15}+\frac{12\!\cdots\!23}{83\!\cdots\!59}a^{13}+\frac{12\!\cdots\!60}{83\!\cdots\!59}a^{11}+\frac{76\!\cdots\!08}{83\!\cdots\!59}a^{9}+\frac{60\!\cdots\!49}{16\!\cdots\!18}a^{7}+\frac{27\!\cdots\!98}{83\!\cdots\!59}a^{5}+\frac{65\!\cdots\!48}{83\!\cdots\!59}a^{3}-\frac{20\!\cdots\!15}{16\!\cdots\!18}a$, $\frac{82\!\cdots\!69}{16\!\cdots\!18}a^{31}+\frac{26\!\cdots\!01}{33\!\cdots\!36}a^{29}+\frac{62\!\cdots\!65}{83\!\cdots\!59}a^{27}+\frac{38\!\cdots\!58}{83\!\cdots\!59}a^{25}+\frac{18\!\cdots\!36}{83\!\cdots\!59}a^{23}+\frac{64\!\cdots\!42}{83\!\cdots\!59}a^{21}+\frac{17\!\cdots\!79}{83\!\cdots\!59}a^{19}+\frac{38\!\cdots\!23}{83\!\cdots\!59}a^{17}+\frac{64\!\cdots\!80}{83\!\cdots\!59}a^{15}+\frac{80\!\cdots\!79}{83\!\cdots\!59}a^{13}+\frac{76\!\cdots\!84}{83\!\cdots\!59}a^{11}+\frac{47\!\cdots\!35}{83\!\cdots\!59}a^{9}+\frac{37\!\cdots\!49}{16\!\cdots\!18}a^{7}+\frac{67\!\cdots\!29}{33\!\cdots\!36}a^{5}+\frac{41\!\cdots\!99}{83\!\cdots\!59}a^{3}-\frac{89\!\cdots\!41}{83\!\cdots\!59}a$, $\frac{19\!\cdots\!61}{15\!\cdots\!58}a^{31}+\frac{62\!\cdots\!41}{30\!\cdots\!16}a^{29}+\frac{29\!\cdots\!65}{15\!\cdots\!58}a^{27}+\frac{37\!\cdots\!41}{30\!\cdots\!16}a^{25}+\frac{44\!\cdots\!86}{75\!\cdots\!79}a^{23}+\frac{15\!\cdots\!42}{75\!\cdots\!79}a^{21}+\frac{44\!\cdots\!92}{75\!\cdots\!79}a^{19}+\frac{96\!\cdots\!98}{75\!\cdots\!79}a^{17}+\frac{16\!\cdots\!40}{75\!\cdots\!79}a^{15}+\frac{21\!\cdots\!09}{75\!\cdots\!79}a^{13}+\frac{20\!\cdots\!04}{75\!\cdots\!79}a^{11}+\frac{13\!\cdots\!70}{75\!\cdots\!79}a^{9}+\frac{12\!\cdots\!13}{15\!\cdots\!58}a^{7}+\frac{48\!\cdots\!13}{30\!\cdots\!16}a^{5}+\frac{35\!\cdots\!23}{15\!\cdots\!58}a^{3}+\frac{25\!\cdots\!13}{30\!\cdots\!16}a+1$, $\frac{10\!\cdots\!65}{16\!\cdots\!18}a^{31}-\frac{43\!\cdots\!41}{16\!\cdots\!18}a^{30}+\frac{33\!\cdots\!65}{33\!\cdots\!36}a^{29}-\frac{13\!\cdots\!55}{33\!\cdots\!36}a^{28}+\frac{31\!\cdots\!25}{33\!\cdots\!36}a^{27}-\frac{33\!\cdots\!00}{83\!\cdots\!59}a^{26}+\frac{49\!\cdots\!81}{83\!\cdots\!59}a^{25}-\frac{41\!\cdots\!85}{16\!\cdots\!18}a^{24}+\frac{23\!\cdots\!65}{83\!\cdots\!59}a^{23}-\frac{98\!\cdots\!00}{83\!\cdots\!59}a^{22}+\frac{81\!\cdots\!30}{83\!\cdots\!59}a^{21}-\frac{35\!\cdots\!38}{83\!\cdots\!59}a^{20}+\frac{22\!\cdots\!80}{83\!\cdots\!59}a^{19}-\frac{97\!\cdots\!80}{83\!\cdots\!59}a^{18}+\frac{48\!\cdots\!45}{83\!\cdots\!59}a^{17}-\frac{21\!\cdots\!80}{83\!\cdots\!59}a^{16}+\frac{81\!\cdots\!00}{83\!\cdots\!59}a^{15}-\frac{36\!\cdots\!80}{83\!\cdots\!59}a^{14}+\frac{10\!\cdots\!85}{83\!\cdots\!59}a^{13}-\frac{46\!\cdots\!60}{83\!\cdots\!59}a^{12}+\frac{96\!\cdots\!10}{83\!\cdots\!59}a^{11}-\frac{45\!\cdots\!12}{83\!\cdots\!59}a^{10}+\frac{60\!\cdots\!25}{83\!\cdots\!59}a^{9}-\frac{30\!\cdots\!20}{83\!\cdots\!59}a^{8}+\frac{48\!\cdots\!95}{16\!\cdots\!18}a^{7}-\frac{27\!\cdots\!85}{16\!\cdots\!18}a^{6}+\frac{86\!\cdots\!85}{33\!\cdots\!36}a^{5}-\frac{10\!\cdots\!75}{33\!\cdots\!36}a^{4}+\frac{20\!\cdots\!65}{33\!\cdots\!36}a^{3}-\frac{40\!\cdots\!10}{83\!\cdots\!59}a^{2}-\frac{97\!\cdots\!70}{83\!\cdots\!59}a-\frac{19\!\cdots\!61}{16\!\cdots\!18}$, $\frac{26\!\cdots\!79}{75\!\cdots\!79}a^{31}-\frac{87589632705}{8243253718082}a^{30}+\frac{16\!\cdots\!03}{30\!\cdots\!16}a^{29}-\frac{13940291661}{82846771036}a^{28}+\frac{39\!\cdots\!35}{75\!\cdots\!79}a^{27}-\frac{6542359560696}{4121626859041}a^{26}+\frac{10\!\cdots\!73}{30\!\cdots\!16}a^{25}-\frac{81919443833915}{8243253718082}a^{24}+\frac{11\!\cdots\!98}{75\!\cdots\!79}a^{23}-\frac{191360253669396}{4121626859041}a^{22}+\frac{42\!\cdots\!11}{75\!\cdots\!79}a^{21}-\frac{676822404992814}{4121626859041}a^{20}+\frac{11\!\cdots\!26}{75\!\cdots\!79}a^{19}-\frac{18\!\cdots\!28}{4121626859041}a^{18}+\frac{25\!\cdots\!34}{75\!\cdots\!79}a^{17}-\frac{40\!\cdots\!53}{4121626859041}a^{16}+\frac{43\!\cdots\!91}{75\!\cdots\!79}a^{15}-\frac{67\!\cdots\!72}{4121626859041}a^{14}+\frac{56\!\cdots\!17}{75\!\cdots\!79}a^{13}-\frac{84\!\cdots\!20}{4121626859041}a^{12}+\frac{55\!\cdots\!42}{75\!\cdots\!79}a^{11}-\frac{79\!\cdots\!16}{4121626859041}a^{10}+\frac{37\!\cdots\!35}{75\!\cdots\!79}a^{9}-\frac{49\!\cdots\!97}{4121626859041}a^{8}+\frac{16\!\cdots\!07}{75\!\cdots\!79}a^{7}-\frac{39\!\cdots\!61}{8243253718082}a^{6}+\frac{13\!\cdots\!59}{30\!\cdots\!16}a^{5}-\frac{709863284825799}{16486507436164}a^{4}+\frac{44\!\cdots\!97}{75\!\cdots\!79}a^{3}-\frac{4298131170078}{4121626859041}a^{2}+\frac{69\!\cdots\!09}{30\!\cdots\!16}a+\frac{13181914833877}{8243253718082}$, $\frac{15\!\cdots\!23}{83\!\cdots\!59}a^{31}+\frac{96\!\cdots\!13}{33\!\cdots\!36}a^{29}+\frac{91\!\cdots\!01}{33\!\cdots\!36}a^{27}+\frac{57\!\cdots\!91}{33\!\cdots\!36}a^{25}+\frac{68\!\cdots\!42}{83\!\cdots\!59}a^{23}+\frac{24\!\cdots\!54}{83\!\cdots\!59}a^{21}+\frac{68\!\cdots\!62}{83\!\cdots\!59}a^{19}+\frac{14\!\cdots\!94}{83\!\cdots\!59}a^{17}+\frac{25\!\cdots\!69}{83\!\cdots\!59}a^{15}+\frac{32\!\cdots\!35}{83\!\cdots\!59}a^{13}+\frac{31\!\cdots\!66}{83\!\cdots\!59}a^{11}+\frac{21\!\cdots\!54}{83\!\cdots\!59}a^{9}+\frac{96\!\cdots\!79}{83\!\cdots\!59}a^{7}+\frac{74\!\cdots\!81}{33\!\cdots\!36}a^{5}+\frac{10\!\cdots\!45}{33\!\cdots\!36}a^{3}+\frac{39\!\cdots\!51}{33\!\cdots\!36}a+1$, $\frac{58\!\cdots\!83}{16\!\cdots\!18}a^{31}-\frac{43\!\cdots\!41}{16\!\cdots\!18}a^{30}+\frac{18\!\cdots\!69}{33\!\cdots\!36}a^{29}-\frac{13\!\cdots\!55}{33\!\cdots\!36}a^{28}+\frac{87\!\cdots\!53}{16\!\cdots\!18}a^{27}-\frac{33\!\cdots\!00}{83\!\cdots\!59}a^{26}+\frac{11\!\cdots\!57}{33\!\cdots\!36}a^{25}-\frac{41\!\cdots\!85}{16\!\cdots\!18}a^{24}+\frac{12\!\cdots\!22}{83\!\cdots\!59}a^{23}-\frac{98\!\cdots\!00}{83\!\cdots\!59}a^{22}+\frac{45\!\cdots\!62}{83\!\cdots\!59}a^{21}-\frac{35\!\cdots\!38}{83\!\cdots\!59}a^{20}+\frac{12\!\cdots\!91}{83\!\cdots\!59}a^{19}-\frac{97\!\cdots\!80}{83\!\cdots\!59}a^{18}+\frac{27\!\cdots\!71}{83\!\cdots\!59}a^{17}-\frac{21\!\cdots\!80}{83\!\cdots\!59}a^{16}+\frac{45\!\cdots\!48}{83\!\cdots\!59}a^{15}-\frac{36\!\cdots\!80}{83\!\cdots\!59}a^{14}+\frac{57\!\cdots\!33}{83\!\cdots\!59}a^{13}-\frac{46\!\cdots\!60}{83\!\cdots\!59}a^{12}+\frac{53\!\cdots\!56}{83\!\cdots\!59}a^{11}-\frac{45\!\cdots\!12}{83\!\cdots\!59}a^{10}+\frac{33\!\cdots\!63}{83\!\cdots\!59}a^{9}-\frac{30\!\cdots\!20}{83\!\cdots\!59}a^{8}+\frac{26\!\cdots\!63}{16\!\cdots\!18}a^{7}-\frac{27\!\cdots\!85}{16\!\cdots\!18}a^{6}+\frac{47\!\cdots\!53}{33\!\cdots\!36}a^{5}-\frac{10\!\cdots\!75}{33\!\cdots\!36}a^{4}+\frac{57\!\cdots\!13}{16\!\cdots\!18}a^{3}-\frac{40\!\cdots\!10}{83\!\cdots\!59}a^{2}-\frac{32\!\cdots\!19}{33\!\cdots\!36}a-\frac{19\!\cdots\!61}{16\!\cdots\!18}$, $\frac{22\!\cdots\!11}{10\!\cdots\!21}a^{31}-\frac{43\!\cdots\!41}{16\!\cdots\!18}a^{30}+\frac{17\!\cdots\!13}{53\!\cdots\!79}a^{29}-\frac{13\!\cdots\!55}{33\!\cdots\!36}a^{28}+\frac{13\!\cdots\!57}{42\!\cdots\!84}a^{27}-\frac{33\!\cdots\!00}{83\!\cdots\!59}a^{26}+\frac{21\!\cdots\!56}{10\!\cdots\!21}a^{25}-\frac{41\!\cdots\!85}{16\!\cdots\!18}a^{24}+\frac{99\!\cdots\!60}{10\!\cdots\!21}a^{23}-\frac{98\!\cdots\!00}{83\!\cdots\!59}a^{22}+\frac{35\!\cdots\!00}{10\!\cdots\!21}a^{21}-\frac{35\!\cdots\!38}{83\!\cdots\!59}a^{20}+\frac{99\!\cdots\!10}{10\!\cdots\!21}a^{19}-\frac{97\!\cdots\!80}{83\!\cdots\!59}a^{18}+\frac{21\!\cdots\!23}{10\!\cdots\!21}a^{17}-\frac{21\!\cdots\!80}{83\!\cdots\!59}a^{16}+\frac{37\!\cdots\!07}{10\!\cdots\!21}a^{15}-\frac{36\!\cdots\!80}{83\!\cdots\!59}a^{14}+\frac{48\!\cdots\!92}{10\!\cdots\!21}a^{13}-\frac{46\!\cdots\!60}{83\!\cdots\!59}a^{12}+\frac{47\!\cdots\!54}{10\!\cdots\!21}a^{11}-\frac{45\!\cdots\!12}{83\!\cdots\!59}a^{10}+\frac{31\!\cdots\!68}{10\!\cdots\!21}a^{9}-\frac{30\!\cdots\!20}{83\!\cdots\!59}a^{8}+\frac{14\!\cdots\!22}{10\!\cdots\!21}a^{7}-\frac{27\!\cdots\!85}{16\!\cdots\!18}a^{6}+\frac{29\!\cdots\!28}{10\!\cdots\!21}a^{5}-\frac{10\!\cdots\!75}{33\!\cdots\!36}a^{4}+\frac{17\!\cdots\!41}{42\!\cdots\!84}a^{3}-\frac{40\!\cdots\!10}{83\!\cdots\!59}a^{2}+\frac{10\!\cdots\!26}{10\!\cdots\!21}a-\frac{28\!\cdots\!43}{16\!\cdots\!18}$, $\frac{77\!\cdots\!08}{83\!\cdots\!59}a^{31}-\frac{70\!\cdots\!95}{33\!\cdots\!36}a^{30}+\frac{49\!\cdots\!89}{33\!\cdots\!36}a^{29}-\frac{11\!\cdots\!95}{33\!\cdots\!36}a^{28}+\frac{47\!\cdots\!59}{33\!\cdots\!36}a^{27}-\frac{26\!\cdots\!69}{83\!\cdots\!59}a^{26}+\frac{29\!\cdots\!79}{33\!\cdots\!36}a^{25}-\frac{65\!\cdots\!97}{33\!\cdots\!36}a^{24}+\frac{35\!\cdots\!14}{83\!\cdots\!59}a^{23}-\frac{76\!\cdots\!88}{83\!\cdots\!59}a^{22}+\frac{12\!\cdots\!78}{83\!\cdots\!59}a^{21}-\frac{26\!\cdots\!94}{83\!\cdots\!59}a^{20}+\frac{35\!\cdots\!78}{83\!\cdots\!59}a^{19}-\frac{74\!\cdots\!96}{83\!\cdots\!59}a^{18}+\frac{76\!\cdots\!22}{83\!\cdots\!59}a^{17}-\frac{15\!\cdots\!09}{83\!\cdots\!59}a^{16}+\frac{13\!\cdots\!92}{83\!\cdots\!59}a^{15}-\frac{26\!\cdots\!08}{83\!\cdots\!59}a^{14}+\frac{16\!\cdots\!51}{83\!\cdots\!59}a^{13}-\frac{33\!\cdots\!53}{83\!\cdots\!59}a^{12}+\frac{16\!\cdots\!10}{83\!\cdots\!59}a^{11}-\frac{31\!\cdots\!40}{83\!\cdots\!59}a^{10}+\frac{11\!\cdots\!54}{83\!\cdots\!59}a^{9}-\frac{19\!\cdots\!53}{83\!\cdots\!59}a^{8}+\frac{50\!\cdots\!36}{83\!\cdots\!59}a^{7}-\frac{31\!\cdots\!51}{33\!\cdots\!36}a^{6}+\frac{38\!\cdots\!37}{33\!\cdots\!36}a^{5}-\frac{28\!\cdots\!27}{33\!\cdots\!36}a^{4}+\frac{52\!\cdots\!71}{33\!\cdots\!36}a^{3}-\frac{17\!\cdots\!23}{83\!\cdots\!59}a^{2}+\frac{20\!\cdots\!99}{33\!\cdots\!36}a+\frac{17\!\cdots\!23}{33\!\cdots\!36}$, $\frac{43\!\cdots\!41}{16\!\cdots\!18}a^{31}+\frac{87589632705}{8243253718082}a^{30}+\frac{13\!\cdots\!55}{33\!\cdots\!36}a^{29}+\frac{13940291661}{82846771036}a^{28}+\frac{33\!\cdots\!00}{83\!\cdots\!59}a^{27}+\frac{6542359560696}{4121626859041}a^{26}+\frac{41\!\cdots\!85}{16\!\cdots\!18}a^{25}+\frac{81919443833915}{8243253718082}a^{24}+\frac{98\!\cdots\!00}{83\!\cdots\!59}a^{23}+\frac{191360253669396}{4121626859041}a^{22}+\frac{35\!\cdots\!38}{83\!\cdots\!59}a^{21}+\frac{676822404992814}{4121626859041}a^{20}+\frac{97\!\cdots\!80}{83\!\cdots\!59}a^{19}+\frac{18\!\cdots\!28}{4121626859041}a^{18}+\frac{21\!\cdots\!80}{83\!\cdots\!59}a^{17}+\frac{40\!\cdots\!53}{4121626859041}a^{16}+\frac{36\!\cdots\!80}{83\!\cdots\!59}a^{15}+\frac{67\!\cdots\!72}{4121626859041}a^{14}+\frac{46\!\cdots\!60}{83\!\cdots\!59}a^{13}+\frac{84\!\cdots\!20}{4121626859041}a^{12}+\frac{45\!\cdots\!12}{83\!\cdots\!59}a^{11}+\frac{79\!\cdots\!16}{4121626859041}a^{10}+\frac{30\!\cdots\!20}{83\!\cdots\!59}a^{9}+\frac{49\!\cdots\!97}{4121626859041}a^{8}+\frac{27\!\cdots\!85}{16\!\cdots\!18}a^{7}+\frac{39\!\cdots\!61}{8243253718082}a^{6}+\frac{10\!\cdots\!75}{33\!\cdots\!36}a^{5}+\frac{709863284825799}{16486507436164}a^{4}+\frac{40\!\cdots\!10}{83\!\cdots\!59}a^{3}+\frac{4298131170078}{4121626859041}a^{2}+\frac{28\!\cdots\!43}{16\!\cdots\!18}a-\frac{13181914833877}{8243253718082}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 48613521256.81357 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 48613521256.81357 \cdot 2560}{6\cdot\sqrt{203282392447840896882957090816000000000000000000000000}}\cr\approx \mathstrut & 0.271439975078757 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times C_4^2$ (as 32T36):
An abelian group of order 32 |
The 32 conjugacy class representatives for $C_2\times C_4^2$ |
Character table for $C_2\times C_4^2$ is not computed |
Intermediate fields
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 | ${\href{/padicField/7.4.0.1}{4} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{16}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{16}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{8}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | Deg $16$ | $8$ | $2$ | $48$ | |||
Deg $16$ | $8$ | $2$ | $48$ | ||||
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(5\) | 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |
5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |