Normalized defining polynomial
\( x^{32} - x^{31} - 3 x^{30} + 7 x^{29} + 5 x^{28} - 33 x^{27} + 13 x^{26} + 119 x^{25} - 171 x^{24} - 305 x^{23} + 989 x^{22} + 231 x^{21} - 4187 x^{20} + 3263 x^{19} + 13485 x^{18} - 26537 x^{17} - 27403 x^{16} - 106148 x^{15} + 215760 x^{14} + 208832 x^{13} - 1071872 x^{12} + 236544 x^{11} + 4050944 x^{10} - 4997120 x^{9} - 11206656 x^{8} + 31195136 x^{7} + 13631488 x^{6} - 138412032 x^{5} + 83886080 x^{4} + 469762048 x^{3} - 805306368 x^{2} - 1073741824 x + 4294967296 \)
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: | \(53818028735688890166227692218037660486426411285400390625=3^{16}\cdot 5^{16}\cdot 17^{30}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $55.16$ | 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: | $3, 5, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(255=3\cdot 5\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{255}(1,·)$, $\chi_{255}(134,·)$, $\chi_{255}(14,·)$, $\chi_{255}(16,·)$, $\chi_{255}(149,·)$, $\chi_{255}(151,·)$, $\chi_{255}(29,·)$, $\chi_{255}(31,·)$, $\chi_{255}(164,·)$, $\chi_{255}(166,·)$, $\chi_{255}(44,·)$, $\chi_{255}(46,·)$, $\chi_{255}(179,·)$, $\chi_{255}(181,·)$, $\chi_{255}(59,·)$, $\chi_{255}(61,·)$, $\chi_{255}(194,·)$, $\chi_{255}(196,·)$, $\chi_{255}(74,·)$, $\chi_{255}(76,·)$, $\chi_{255}(209,·)$, $\chi_{255}(211,·)$, $\chi_{255}(89,·)$, $\chi_{255}(91,·)$, $\chi_{255}(224,·)$, $\chi_{255}(226,·)$, $\chi_{255}(104,·)$, $\chi_{255}(106,·)$, $\chi_{255}(239,·)$, $\chi_{255}(241,·)$, $\chi_{255}(121,·)$, $\chi_{255}(254,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{109612} a^{17} + \frac{1}{4} a^{16} - \frac{1}{4} a^{15} + \frac{1}{4} a^{14} - \frac{1}{4} a^{13} + \frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{866}{27403}$, $\frac{1}{438448} a^{18} - \frac{1}{438448} a^{17} + \frac{7}{16} a^{16} + \frac{5}{16} a^{15} - \frac{1}{16} a^{14} - \frac{3}{16} a^{13} + \frac{7}{16} a^{12} + \frac{5}{16} a^{11} - \frac{1}{16} a^{10} - \frac{3}{16} a^{9} + \frac{7}{16} a^{8} + \frac{5}{16} a^{7} - \frac{1}{16} a^{6} - \frac{3}{16} a^{5} + \frac{7}{16} a^{4} + \frac{5}{16} a^{3} - \frac{1}{16} a^{2} - \frac{26537}{109612} a + \frac{13485}{27403}$, $\frac{1}{1753792} a^{19} - \frac{1}{1753792} a^{18} - \frac{3}{1753792} a^{17} - \frac{11}{64} a^{16} - \frac{17}{64} a^{15} - \frac{3}{64} a^{14} + \frac{7}{64} a^{13} + \frac{5}{64} a^{12} + \frac{31}{64} a^{11} + \frac{13}{64} a^{10} - \frac{9}{64} a^{9} + \frac{21}{64} a^{8} + \frac{15}{64} a^{7} + \frac{29}{64} a^{6} - \frac{25}{64} a^{5} - \frac{27}{64} a^{4} - \frac{1}{64} a^{3} - \frac{26537}{438448} a^{2} + \frac{13485}{109612} a + \frac{3263}{27403}$, $\frac{1}{7015168} a^{20} - \frac{1}{7015168} a^{19} - \frac{3}{7015168} a^{18} + \frac{7}{7015168} a^{17} + \frac{47}{256} a^{16} - \frac{3}{256} a^{15} + \frac{71}{256} a^{14} - \frac{59}{256} a^{13} + \frac{31}{256} a^{12} - \frac{51}{256} a^{11} - \frac{73}{256} a^{10} + \frac{21}{256} a^{9} + \frac{15}{256} a^{8} - \frac{99}{256} a^{7} + \frac{39}{256} a^{6} + \frac{101}{256} a^{5} - \frac{1}{256} a^{4} - \frac{26537}{1753792} a^{3} + \frac{13485}{438448} a^{2} + \frac{3263}{109612} a - \frac{4187}{27403}$, $\frac{1}{28060672} a^{21} - \frac{1}{28060672} a^{20} - \frac{3}{28060672} a^{19} + \frac{7}{28060672} a^{18} + \frac{5}{28060672} a^{17} - \frac{259}{1024} a^{16} + \frac{71}{1024} a^{15} - \frac{59}{1024} a^{14} - \frac{225}{1024} a^{13} + \frac{461}{1024} a^{12} + \frac{439}{1024} a^{11} - \frac{235}{1024} a^{10} - \frac{497}{1024} a^{9} + \frac{413}{1024} a^{8} - \frac{473}{1024} a^{7} - \frac{155}{1024} a^{6} - \frac{1}{1024} a^{5} - \frac{26537}{7015168} a^{4} + \frac{13485}{1753792} a^{3} + \frac{3263}{438448} a^{2} - \frac{4187}{109612} a + \frac{231}{27403}$, $\frac{1}{112242688} a^{22} - \frac{1}{112242688} a^{21} - \frac{3}{112242688} a^{20} + \frac{7}{112242688} a^{19} + \frac{5}{112242688} a^{18} - \frac{33}{112242688} a^{17} + \frac{71}{4096} a^{16} + \frac{965}{4096} a^{15} - \frac{1249}{4096} a^{14} + \frac{1485}{4096} a^{13} - \frac{585}{4096} a^{12} - \frac{1259}{4096} a^{11} - \frac{497}{4096} a^{10} + \frac{1437}{4096} a^{9} + \frac{551}{4096} a^{8} + \frac{1893}{4096} a^{7} - \frac{1}{4096} a^{6} - \frac{26537}{28060672} a^{5} + \frac{13485}{7015168} a^{4} + \frac{3263}{1753792} a^{3} - \frac{4187}{438448} a^{2} + \frac{231}{109612} a + \frac{989}{27403}$, $\frac{1}{448970752} a^{23} - \frac{1}{448970752} a^{22} - \frac{3}{448970752} a^{21} + \frac{7}{448970752} a^{20} + \frac{5}{448970752} a^{19} - \frac{33}{448970752} a^{18} + \frac{13}{448970752} a^{17} + \frac{965}{16384} a^{16} - \frac{1249}{16384} a^{15} - \frac{2611}{16384} a^{14} + \frac{7607}{16384} a^{13} + \frac{2837}{16384} a^{12} - \frac{497}{16384} a^{11} + \frac{5533}{16384} a^{10} - \frac{3545}{16384} a^{9} - \frac{2203}{16384} a^{8} - \frac{1}{16384} a^{7} - \frac{26537}{112242688} a^{6} + \frac{13485}{28060672} a^{5} + \frac{3263}{7015168} a^{4} - \frac{4187}{1753792} a^{3} + \frac{231}{438448} a^{2} + \frac{989}{109612} a - \frac{305}{27403}$, $\frac{1}{1795883008} a^{24} - \frac{1}{1795883008} a^{23} - \frac{3}{1795883008} a^{22} + \frac{7}{1795883008} a^{21} + \frac{5}{1795883008} a^{20} - \frac{33}{1795883008} a^{19} + \frac{13}{1795883008} a^{18} + \frac{119}{1795883008} a^{17} - \frac{17633}{65536} a^{16} + \frac{13773}{65536} a^{15} - \frac{8777}{65536} a^{14} + \frac{19221}{65536} a^{13} + \frac{15887}{65536} a^{12} - \frac{27235}{65536} a^{11} + \frac{29223}{65536} a^{10} + \frac{14181}{65536} a^{9} - \frac{1}{65536} a^{8} - \frac{26537}{448970752} a^{7} + \frac{13485}{112242688} a^{6} + \frac{3263}{28060672} a^{5} - \frac{4187}{7015168} a^{4} + \frac{231}{1753792} a^{3} + \frac{989}{438448} a^{2} - \frac{305}{109612} a - \frac{171}{27403}$, $\frac{1}{7183532032} a^{25} - \frac{1}{7183532032} a^{24} - \frac{3}{7183532032} a^{23} + \frac{7}{7183532032} a^{22} + \frac{5}{7183532032} a^{21} - \frac{33}{7183532032} a^{20} + \frac{13}{7183532032} a^{19} + \frac{119}{7183532032} a^{18} - \frac{171}{7183532032} a^{17} - \frac{51763}{262144} a^{16} + \frac{122295}{262144} a^{15} + \frac{84757}{262144} a^{14} - \frac{49649}{262144} a^{13} - \frac{27235}{262144} a^{12} - \frac{36313}{262144} a^{11} - \frac{116891}{262144} a^{10} - \frac{1}{262144} a^{9} - \frac{26537}{1795883008} a^{8} + \frac{13485}{448970752} a^{7} + \frac{3263}{112242688} a^{6} - \frac{4187}{28060672} a^{5} + \frac{231}{7015168} a^{4} + \frac{989}{1753792} a^{3} - \frac{305}{438448} a^{2} - \frac{171}{109612} a + \frac{119}{27403}$, $\frac{1}{28734128128} a^{26} - \frac{1}{28734128128} a^{25} - \frac{3}{28734128128} a^{24} + \frac{7}{28734128128} a^{23} + \frac{5}{28734128128} a^{22} - \frac{33}{28734128128} a^{21} + \frac{13}{28734128128} a^{20} + \frac{119}{28734128128} a^{19} - \frac{171}{28734128128} a^{18} - \frac{305}{28734128128} a^{17} + \frac{122295}{1048576} a^{16} + \frac{84757}{1048576} a^{15} + \frac{474639}{1048576} a^{14} + \frac{234909}{1048576} a^{13} - \frac{36313}{1048576} a^{12} + \frac{145253}{1048576} a^{11} - \frac{1}{1048576} a^{10} - \frac{26537}{7183532032} a^{9} + \frac{13485}{1795883008} a^{8} + \frac{3263}{448970752} a^{7} - \frac{4187}{112242688} a^{6} + \frac{231}{28060672} a^{5} + \frac{989}{7015168} a^{4} - \frac{305}{1753792} a^{3} - \frac{171}{438448} a^{2} + \frac{119}{109612} a + \frac{13}{27403}$, $\frac{1}{114936512512} a^{27} - \frac{1}{114936512512} a^{26} - \frac{3}{114936512512} a^{25} + \frac{7}{114936512512} a^{24} + \frac{5}{114936512512} a^{23} - \frac{33}{114936512512} a^{22} + \frac{13}{114936512512} a^{21} + \frac{119}{114936512512} a^{20} - \frac{171}{114936512512} a^{19} - \frac{305}{114936512512} a^{18} + \frac{989}{114936512512} a^{17} - \frac{963819}{4194304} a^{16} + \frac{474639}{4194304} a^{15} - \frac{813667}{4194304} a^{14} - \frac{1084889}{4194304} a^{13} + \frac{145253}{4194304} a^{12} - \frac{1}{4194304} a^{11} - \frac{26537}{28734128128} a^{10} + \frac{13485}{7183532032} a^{9} + \frac{3263}{1795883008} a^{8} - \frac{4187}{448970752} a^{7} + \frac{231}{112242688} a^{6} + \frac{989}{28060672} a^{5} - \frac{305}{7015168} a^{4} - \frac{171}{1753792} a^{3} + \frac{119}{438448} a^{2} + \frac{13}{109612} a - \frac{33}{27403}$, $\frac{1}{459746050048} a^{28} - \frac{1}{459746050048} a^{27} - \frac{3}{459746050048} a^{26} + \frac{7}{459746050048} a^{25} + \frac{5}{459746050048} a^{24} - \frac{33}{459746050048} a^{23} + \frac{13}{459746050048} a^{22} + \frac{119}{459746050048} a^{21} - \frac{171}{459746050048} a^{20} - \frac{305}{459746050048} a^{19} + \frac{989}{459746050048} a^{18} + \frac{231}{459746050048} a^{17} + \frac{4668943}{16777216} a^{16} - \frac{813667}{16777216} a^{15} - \frac{1084889}{16777216} a^{14} + \frac{4339557}{16777216} a^{13} - \frac{1}{16777216} a^{12} - \frac{26537}{114936512512} a^{11} + \frac{13485}{28734128128} a^{10} + \frac{3263}{7183532032} a^{9} - \frac{4187}{1795883008} a^{8} + \frac{231}{448970752} a^{7} + \frac{989}{112242688} a^{6} - \frac{305}{28060672} a^{5} - \frac{171}{7015168} a^{4} + \frac{119}{1753792} a^{3} + \frac{13}{438448} a^{2} - \frac{33}{109612} a + \frac{5}{27403}$, $\frac{1}{1838984200192} a^{29} - \frac{1}{1838984200192} a^{28} - \frac{3}{1838984200192} a^{27} + \frac{7}{1838984200192} a^{26} + \frac{5}{1838984200192} a^{25} - \frac{33}{1838984200192} a^{24} + \frac{13}{1838984200192} a^{23} + \frac{119}{1838984200192} a^{22} - \frac{171}{1838984200192} a^{21} - \frac{305}{1838984200192} a^{20} + \frac{989}{1838984200192} a^{19} + \frac{231}{1838984200192} a^{18} - \frac{4187}{1838984200192} a^{17} - \frac{17590883}{67108864} a^{16} - \frac{1084889}{67108864} a^{15} + \frac{4339557}{67108864} a^{14} - \frac{1}{67108864} a^{13} - \frac{26537}{459746050048} a^{12} + \frac{13485}{114936512512} a^{11} + \frac{3263}{28734128128} a^{10} - \frac{4187}{7183532032} a^{9} + \frac{231}{1795883008} a^{8} + \frac{989}{448970752} a^{7} - \frac{305}{112242688} a^{6} - \frac{171}{28060672} a^{5} + \frac{119}{7015168} a^{4} + \frac{13}{1753792} a^{3} - \frac{33}{438448} a^{2} + \frac{5}{109612} a + \frac{7}{27403}$, $\frac{1}{7355936800768} a^{30} - \frac{1}{7355936800768} a^{29} - \frac{3}{7355936800768} a^{28} + \frac{7}{7355936800768} a^{27} + \frac{5}{7355936800768} a^{26} - \frac{33}{7355936800768} a^{25} + \frac{13}{7355936800768} a^{24} + \frac{119}{7355936800768} a^{23} - \frac{171}{7355936800768} a^{22} - \frac{305}{7355936800768} a^{21} + \frac{989}{7355936800768} a^{20} + \frac{231}{7355936800768} a^{19} - \frac{4187}{7355936800768} a^{18} + \frac{3263}{7355936800768} a^{17} + \frac{66023975}{268435456} a^{16} + \frac{4339557}{268435456} a^{15} - \frac{1}{268435456} a^{14} - \frac{26537}{1838984200192} a^{13} + \frac{13485}{459746050048} a^{12} + \frac{3263}{114936512512} a^{11} - \frac{4187}{28734128128} a^{10} + \frac{231}{7183532032} a^{9} + \frac{989}{1795883008} a^{8} - \frac{305}{448970752} a^{7} - \frac{171}{112242688} a^{6} + \frac{119}{28060672} a^{5} + \frac{13}{7015168} a^{4} - \frac{33}{1753792} a^{3} + \frac{5}{438448} a^{2} + \frac{7}{109612} a - \frac{3}{27403}$, $\frac{1}{29423747203072} a^{31} - \frac{1}{29423747203072} a^{30} - \frac{3}{29423747203072} a^{29} + \frac{7}{29423747203072} a^{28} + \frac{5}{29423747203072} a^{27} - \frac{33}{29423747203072} a^{26} + \frac{13}{29423747203072} a^{25} + \frac{119}{29423747203072} a^{24} - \frac{171}{29423747203072} a^{23} - \frac{305}{29423747203072} a^{22} + \frac{989}{29423747203072} a^{21} + \frac{231}{29423747203072} a^{20} - \frac{4187}{29423747203072} a^{19} + \frac{3263}{29423747203072} a^{18} + \frac{13485}{29423747203072} a^{17} - \frac{264095899}{1073741824} a^{16} - \frac{1}{1073741824} a^{15} - \frac{26537}{7355936800768} a^{14} + \frac{13485}{1838984200192} a^{13} + \frac{3263}{459746050048} a^{12} - \frac{4187}{114936512512} a^{11} + \frac{231}{28734128128} a^{10} + \frac{989}{7183532032} a^{9} - \frac{305}{1795883008} a^{8} - \frac{171}{448970752} a^{7} + \frac{119}{112242688} a^{6} + \frac{13}{28060672} a^{5} - \frac{33}{7015168} a^{4} + \frac{5}{1753792} a^{3} + \frac{7}{438448} a^{2} - \frac{3}{109612} a - \frac{1}{27403}$
Class group and class number
Not computed
Unit group
| Rank: | $15$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{13485}{29423747203072} a^{31} - \frac{40455}{29423747203072} a^{30} + \frac{94395}{29423747203072} a^{29} + \frac{67425}{29423747203072} a^{28} - \frac{445005}{29423747203072} a^{27} + \frac{175305}{29423747203072} a^{26} + \frac{1604715}{29423747203072} a^{25} - \frac{2305935}{29423747203072} a^{24} - \frac{4112925}{29423747203072} a^{23} + \frac{13336665}{29423747203072} a^{22} + \frac{3115035}{29423747203072} a^{21} - \frac{56461695}{29423747203072} a^{20} + \frac{44001555}{29423747203072} a^{19} + \frac{181845225}{29423747203072} a^{18} - \frac{357851445}{29423747203072} a^{17} - \frac{13485}{1073741824} a^{16} + \frac{26537}{1073741824} a^{15} + \frac{181845225}{1838984200192} a^{14} + \frac{44001555}{459746050048} a^{13} - \frac{56461695}{114936512512} a^{12} + \frac{3115035}{28734128128} a^{11} + \frac{13336665}{7183532032} a^{10} - \frac{4112925}{1795883008} a^{9} - \frac{2305935}{448970752} a^{8} + \frac{1604715}{112242688} a^{7} + \frac{175305}{28060672} a^{6} - \frac{445005}{7015168} a^{5} + \frac{67425}{1753792} a^{4} + \frac{94395}{438448} a^{3} - \frac{40455}{109612} a^{2} - \frac{13485}{27403} a + \frac{53940}{27403} \) (order $34$) | 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
$C_2\times C_{16}$ (as 32T32):
| An abelian group of order 32 |
| The 32 conjugacy class representatives for $C_2\times C_{16}$ |
| Character table for $C_2\times C_{16}$ 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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{4}$ | R | R | $16^{2}$ | $16^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ | R | ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| 17 | Data not computed | ||||||