Normalized defining polynomial
\( x^{32} - 4 x^{31} + 10 x^{30} - 20 x^{29} + 34 x^{28} - 32 x^{27} - 8 x^{26} + 112 x^{25} - 308 x^{24} + 608 x^{23} - 880 x^{22} + 944 x^{21} - 536 x^{20} - 672 x^{19} + 2960 x^{18} + \cdots + 65536 \)
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: | \(7669926418924454281216000000000000000000000000\) \(\medspace = 2^{64}\cdot 5^{24}\cdot 17^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(27.16\) | 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: | $2^{2}5^{3/4}17^{1/2}\approx 55.14573827051111$ | ||
Ramified primes: | \(2\), \(5\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | unavailable$^{32768}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{8}a^{12}$, $\frac{1}{8}a^{13}$, $\frac{1}{8}a^{14}$, $\frac{1}{8}a^{15}$, $\frac{1}{16}a^{16}$, $\frac{1}{16}a^{17}$, $\frac{1}{32}a^{18}-\frac{1}{16}a^{14}-\frac{1}{8}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{64}a^{19}-\frac{1}{32}a^{17}+\frac{1}{32}a^{15}-\frac{1}{16}a^{11}+\frac{1}{8}a^{7}-\frac{1}{4}a^{5}+\frac{1}{4}a^{3}$, $\frac{1}{128}a^{20}-\frac{1}{64}a^{18}-\frac{1}{32}a^{17}+\frac{1}{64}a^{16}-\frac{1}{16}a^{15}-\frac{1}{32}a^{12}-\frac{1}{8}a^{11}-\frac{1}{8}a^{9}+\frac{1}{16}a^{8}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}$, $\frac{1}{128}a^{21}-\frac{1}{64}a^{17}+\frac{1}{32}a^{15}-\frac{1}{16}a^{14}-\frac{1}{32}a^{13}-\frac{1}{16}a^{11}+\frac{1}{16}a^{9}-\frac{1}{8}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{256}a^{22}-\frac{1}{128}a^{18}-\frac{1}{32}a^{17}+\frac{1}{64}a^{16}-\frac{1}{32}a^{15}-\frac{1}{64}a^{14}-\frac{1}{16}a^{13}-\frac{1}{32}a^{12}-\frac{1}{8}a^{11}+\frac{1}{32}a^{10}-\frac{1}{8}a^{9}-\frac{1}{16}a^{6}+\frac{1}{8}a^{4}-\frac{1}{4}a^{3}$, $\frac{1}{512}a^{23}-\frac{1}{256}a^{21}+\frac{1}{256}a^{19}+\frac{1}{64}a^{16}+\frac{7}{128}a^{15}+\frac{1}{32}a^{14}+\frac{1}{64}a^{11}-\frac{1}{8}a^{10}-\frac{1}{32}a^{9}+\frac{1}{32}a^{7}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{8}a^{4}$, $\frac{1}{1024}a^{24}-\frac{1}{512}a^{22}-\frac{1}{256}a^{21}+\frac{1}{512}a^{20}-\frac{1}{128}a^{19}-\frac{1}{64}a^{18}-\frac{1}{256}a^{16}+\frac{3}{64}a^{15}+\frac{1}{64}a^{13}-\frac{7}{128}a^{12}-\frac{5}{64}a^{10}-\frac{1}{32}a^{9}+\frac{1}{64}a^{8}+\frac{1}{8}a^{7}-\frac{1}{4}a^{6}+\frac{1}{8}a^{5}+\frac{1}{4}a^{3}$, $\frac{1}{1024}a^{25}-\frac{1}{512}a^{21}+\frac{1}{256}a^{19}+\frac{1}{128}a^{18}+\frac{7}{256}a^{17}-\frac{1}{32}a^{16}-\frac{1}{128}a^{15}-\frac{1}{32}a^{14}+\frac{1}{128}a^{13}-\frac{1}{16}a^{12}-\frac{1}{8}a^{11}-\frac{1}{64}a^{9}-\frac{1}{16}a^{8}-\frac{3}{32}a^{7}+\frac{1}{16}a^{6}-\frac{1}{4}a^{5}-\frac{1}{8}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2048}a^{26}-\frac{1}{1024}a^{22}-\frac{1}{256}a^{21}+\frac{1}{512}a^{20}-\frac{1}{256}a^{19}-\frac{1}{512}a^{18}-\frac{3}{128}a^{17}+\frac{7}{256}a^{16}+\frac{1}{64}a^{15}+\frac{1}{256}a^{14}-\frac{1}{64}a^{13}+\frac{1}{16}a^{11}-\frac{9}{128}a^{10}-\frac{1}{16}a^{9}-\frac{3}{64}a^{8}+\frac{7}{32}a^{7}+\frac{1}{8}a^{5}$, $\frac{1}{4096}a^{27}-\frac{1}{2048}a^{25}+\frac{1}{2048}a^{23}+\frac{1}{512}a^{20}+\frac{7}{1024}a^{19}-\frac{3}{256}a^{18}+\frac{1}{64}a^{17}-\frac{1}{64}a^{16}-\frac{31}{512}a^{15}-\frac{1}{64}a^{14}+\frac{7}{256}a^{13}+\frac{1}{32}a^{12}-\frac{31}{256}a^{11}-\frac{1}{64}a^{10}+\frac{3}{32}a^{9}+\frac{3}{64}a^{8}-\frac{1}{8}a^{7}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{8192}a^{28}-\frac{1}{4096}a^{26}-\frac{1}{2048}a^{25}+\frac{1}{4096}a^{24}-\frac{1}{1024}a^{23}-\frac{1}{512}a^{22}-\frac{1}{2048}a^{20}+\frac{3}{512}a^{19}-\frac{1}{64}a^{18}-\frac{7}{512}a^{17}-\frac{7}{1024}a^{16}+\frac{1}{32}a^{15}-\frac{5}{512}a^{14}-\frac{1}{256}a^{13}-\frac{31}{512}a^{12}-\frac{7}{64}a^{11}+\frac{3}{32}a^{10}-\frac{7}{64}a^{9}+\frac{1}{16}a^{8}-\frac{3}{32}a^{7}-\frac{1}{8}a^{6}-\frac{1}{8}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{8192}a^{29}-\frac{1}{4096}a^{25}+\frac{1}{2048}a^{23}+\frac{1}{1024}a^{22}+\frac{7}{2048}a^{21}-\frac{1}{256}a^{20}-\frac{1}{1024}a^{19}+\frac{3}{256}a^{18}+\frac{17}{1024}a^{17}-\frac{3}{128}a^{16}-\frac{1}{64}a^{15}-\frac{1}{16}a^{14}-\frac{17}{512}a^{13}+\frac{3}{128}a^{12}-\frac{3}{256}a^{11}+\frac{1}{128}a^{10}+\frac{1}{32}a^{9}+\frac{3}{64}a^{8}-\frac{1}{4}a^{7}-\frac{1}{16}a^{6}+\frac{1}{8}a^{4}+\frac{1}{4}a^{3}$, $\frac{1}{16384}a^{30}-\frac{1}{8192}a^{26}-\frac{1}{2048}a^{25}+\frac{1}{4096}a^{24}-\frac{1}{2048}a^{23}-\frac{1}{4096}a^{22}-\frac{3}{1024}a^{21}+\frac{7}{2048}a^{20}+\frac{1}{512}a^{19}+\frac{1}{2048}a^{18}-\frac{9}{512}a^{17}-\frac{1}{32}a^{16}+\frac{5}{128}a^{15}-\frac{41}{1024}a^{14}+\frac{7}{128}a^{13}-\frac{19}{512}a^{12}+\frac{23}{256}a^{11}+\frac{1}{16}a^{10}-\frac{7}{64}a^{9}+\frac{1}{16}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}$, $\frac{1}{32768}a^{31}-\frac{1}{16384}a^{29}+\frac{1}{16384}a^{27}+\frac{1}{4096}a^{24}+\frac{7}{8192}a^{23}-\frac{3}{2048}a^{22}+\frac{1}{512}a^{21}-\frac{1}{512}a^{20}-\frac{31}{4096}a^{19}-\frac{1}{512}a^{18}-\frac{25}{2048}a^{17}+\frac{1}{256}a^{16}-\frac{95}{2048}a^{15}-\frac{17}{512}a^{14}+\frac{3}{256}a^{13}-\frac{13}{512}a^{12}+\frac{3}{64}a^{11}-\frac{1}{8}a^{10}+\frac{3}{64}a^{8}+\frac{1}{16}a^{7}-\frac{1}{16}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{5}$, which has order $5$ (assuming GRH)
Unit group
Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{1}{512} a^{31} - \frac{13}{4096} a^{30} + \frac{9}{2048} a^{29} - \frac{1}{64} a^{28} + \frac{17}{512} a^{27} - \frac{145}{2048} a^{26} + \frac{7}{1024} a^{25} + \frac{15}{512} a^{24} - \frac{1}{4} a^{23} + \frac{609}{1024} a^{22} - \frac{531}{512} a^{21} + \frac{281}{256} a^{20} - \frac{255}{256} a^{19} - \frac{143}{512} a^{18} + \frac{697}{256} a^{17} - \frac{1629}{256} a^{16} + \frac{685}{64} a^{15} - \frac{3681}{256} a^{14} + \frac{1927}{128} a^{13} - \frac{601}{64} a^{12} - \frac{35}{4} a^{11} + \frac{963}{32} a^{10} - \frac{545}{8} a^{9} + \frac{6241}{64} a^{8} - \frac{213}{2} a^{7} + \frac{229}{4} a^{6} - 40 a^{5} - 110 a^{4} + \frac{739}{4} a^{3} - 226 a^{2} + 168 a - 232 \) (order $40$) | 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: | $\frac{5}{1024}a^{31}-\frac{43}{4096}a^{30}+\frac{53}{2048}a^{29}-\frac{3}{64}a^{28}+\frac{33}{512}a^{27}-\frac{15}{2048}a^{26}-\frac{93}{1024}a^{25}+\frac{189}{512}a^{24}-\frac{201}{256}a^{23}+\frac{1271}{1024}a^{22}-\frac{711}{512}a^{21}+\frac{279}{256}a^{20}+\frac{109}{256}a^{19}-\frac{1713}{512}a^{18}+\frac{1957}{256}a^{17}-\frac{3251}{256}a^{16}+\frac{1081}{64}a^{15}-\frac{4463}{256}a^{14}+\frac{1339}{128}a^{13}+\frac{521}{64}a^{12}-\frac{139}{4}a^{11}+\frac{2349}{32}a^{10}-\frac{827}{8}a^{9}+\frac{7055}{64}a^{8}-\frac{147}{2}a^{7}+\frac{107}{4}a^{6}+100a^{5}-178a^{4}+216a^{3}-190a^{2}+184a-23$, $\frac{65}{16384}a^{31}-\frac{161}{16384}a^{30}+\frac{161}{8192}a^{29}-\frac{161}{4096}a^{28}+\frac{483}{8192}a^{27}-\frac{123}{8192}a^{26}-\frac{161}{2048}a^{25}+\frac{1127}{4096}a^{24}-\frac{2737}{4096}a^{23}+\frac{4669}{4096}a^{22}-\frac{349}{256}a^{21}+\frac{2415}{2048}a^{20}-\frac{161}{2048}a^{19}-\frac{5313}{2048}a^{18}+\frac{6923}{1024}a^{17}-\frac{6129}{512}a^{16}+\frac{17227}{1024}a^{15}-\frac{19159}{1024}a^{14}+\frac{1771}{128}a^{13}+\frac{1127}{512}a^{12}-\frac{443}{16}a^{11}+\frac{2093}{32}a^{10}-\frac{3381}{32}a^{9}+\frac{7889}{64}a^{8}-\frac{805}{8}a^{7}+\frac{1033}{16}a^{6}+\frac{161}{4}a^{5}-\frac{1449}{8}a^{4}+\frac{483}{2}a^{3}-\frac{483}{2}a^{2}+264a-162$, $\frac{15}{16384}a^{31}-\frac{11}{16384}a^{30}+\frac{51}{8192}a^{29}-\frac{31}{4096}a^{28}+\frac{45}{8192}a^{27}+\frac{63}{8192}a^{26}-\frac{25}{2048}a^{25}+\frac{385}{4096}a^{24}-\frac{479}{4096}a^{23}+\frac{415}{4096}a^{22}-\frac{13}{512}a^{21}-\frac{183}{2048}a^{20}+\frac{1033}{2048}a^{19}-\frac{1539}{2048}a^{18}+\frac{905}{1024}a^{17}-\frac{373}{512}a^{16}+\frac{69}{1024}a^{15}+\frac{1307}{1024}a^{14}-\frac{27}{8}a^{13}+\frac{3041}{512}a^{12}-\frac{113}{16}a^{11}+8a^{10}+\frac{73}{32}a^{9}-\frac{417}{32}a^{8}+\frac{217}{8}a^{7}-\frac{605}{16}a^{6}+\frac{239}{4}a^{5}+\frac{25}{8}a^{4}-\frac{51}{2}a^{3}+\frac{103}{2}a^{2}-80a+137$, $\frac{17}{16384}a^{31}-\frac{3}{4096}a^{30}+\frac{5}{4096}a^{29}+\frac{33}{8192}a^{28}-\frac{101}{8192}a^{27}+\frac{147}{4096}a^{26}-\frac{175}{4096}a^{25}+\frac{211}{4096}a^{24}+\frac{99}{4096}a^{23}-\frac{21}{128}a^{22}+\frac{837}{2048}a^{21}-\frac{1305}{2048}a^{20}+\frac{1709}{2048}a^{19}-\frac{81}{128}a^{18}-\frac{19}{256}a^{17}+\frac{1505}{1024}a^{16}-\frac{3661}{1024}a^{15}+\frac{3133}{512}a^{14}-\frac{4267}{512}a^{13}+\frac{4539}{512}a^{12}-\frac{143}{32}a^{11}-\frac{125}{32}a^{10}+\frac{1313}{64}a^{9}-\frac{293}{8}a^{8}+\frac{1649}{32}a^{7}-\frac{175}{4}a^{6}+\frac{245}{8}a^{5}+\frac{203}{8}a^{4}-\frac{113}{2}a^{3}+\frac{183}{2}a^{2}-83a+80$, $\frac{51}{8192}a^{31}-\frac{231}{16384}a^{30}+\frac{231}{8192}a^{29}-\frac{423}{8192}a^{28}+\frac{295}{4096}a^{27}+\frac{49}{8192}a^{26}-\frac{581}{4096}a^{25}+\frac{869}{2048}a^{24}-\frac{1869}{2048}a^{23}+\frac{5959}{4096}a^{22}-\frac{3231}{2048}a^{21}+\frac{1163}{1024}a^{20}+\frac{289}{512}a^{19}-\frac{8443}{2048}a^{18}+\frac{9579}{1024}a^{17}-\frac{15855}{1024}a^{16}+\frac{10519}{512}a^{15}-\frac{21639}{1024}a^{14}+\frac{6433}{512}a^{13}+\frac{2605}{256}a^{12}-\frac{1353}{32}a^{11}+\frac{1409}{16}a^{10}-\frac{8359}{64}a^{9}+\frac{9073}{64}a^{8}-\frac{3155}{32}a^{7}+\frac{837}{16}a^{6}+\frac{663}{8}a^{5}-\frac{1853}{8}a^{4}+289a^{3}-262a^{2}+293a-153$, $\frac{1}{512}a^{31}+\frac{13}{4096}a^{30}-\frac{9}{2048}a^{29}+\frac{1}{64}a^{28}-\frac{17}{512}a^{27}+\frac{145}{2048}a^{26}-\frac{7}{1024}a^{25}-\frac{15}{512}a^{24}+\frac{1}{4}a^{23}-\frac{609}{1024}a^{22}+\frac{531}{512}a^{21}-\frac{281}{256}a^{20}+\frac{255}{256}a^{19}+\frac{143}{512}a^{18}-\frac{697}{256}a^{17}+\frac{1629}{256}a^{16}-\frac{685}{64}a^{15}+\frac{3681}{256}a^{14}-\frac{1927}{128}a^{13}+\frac{601}{64}a^{12}+\frac{35}{4}a^{11}-\frac{963}{32}a^{10}+\frac{545}{8}a^{9}-\frac{6241}{64}a^{8}+\frac{213}{2}a^{7}-\frac{229}{4}a^{6}+40a^{5}+110a^{4}-\frac{739}{4}a^{3}+226a^{2}-168a+231$, $\frac{147}{16384}a^{31}-\frac{505}{16384}a^{30}+\frac{505}{8192}a^{29}-\frac{505}{4096}a^{28}+\frac{1515}{8192}a^{27}-\frac{763}{8192}a^{26}-\frac{505}{2048}a^{25}+\frac{3535}{4096}a^{24}-\frac{8585}{4096}a^{23}+\frac{14645}{4096}a^{22}-\frac{4533}{1024}a^{21}+\frac{7575}{2048}a^{20}-\frac{505}{2048}a^{19}-\frac{16665}{2048}a^{18}+\frac{21715}{1024}a^{17}-\frac{19219}{512}a^{16}+\frac{54035}{1024}a^{15}-\frac{60095}{1024}a^{14}+\frac{5555}{128}a^{13}+\frac{3535}{512}a^{12}-\frac{23411}{256}a^{11}+\frac{6565}{32}a^{10}-\frac{10605}{32}a^{9}+\frac{24745}{64}a^{8}-\frac{2525}{8}a^{7}+155a^{6}+\frac{505}{4}a^{5}-\frac{4545}{8}a^{4}+\frac{1515}{2}a^{3}-\frac{1515}{2}a^{2}+708a-506$, $\frac{1}{4096}a^{31}+\frac{51}{16384}a^{30}-\frac{49}{8192}a^{29}+\frac{123}{8192}a^{28}-\frac{53}{2048}a^{27}+\frac{295}{8192}a^{26}+\frac{41}{4096}a^{25}-\frac{141}{2048}a^{24}+\frac{31}{128}a^{23}-\frac{1951}{4096}a^{22}+\frac{1467}{2048}a^{21}-\frac{719}{1024}a^{20}+\frac{439}{1024}a^{19}+\frac{1291}{2048}a^{18}-\frac{2529}{1024}a^{17}+\frac{5131}{1024}a^{16}-\frac{1989}{256}a^{15}+\frac{9895}{1024}a^{14}-\frac{4589}{512}a^{13}+\frac{879}{256}a^{12}+\frac{151}{16}a^{11}-\frac{3285}{128}a^{10}+\frac{1563}{32}a^{9}-\frac{4083}{64}a^{8}+\frac{507}{8}a^{7}-\frac{515}{16}a^{6}+5a^{5}+\frac{161}{2}a^{4}-\frac{241}{2}a^{3}+\frac{279}{2}a^{2}-110a+118$, $\frac{41}{8192}a^{31}-\frac{259}{16384}a^{30}+\frac{261}{8192}a^{29}-\frac{281}{4096}a^{28}+\frac{439}{4096}a^{27}-\frac{533}{8192}a^{26}-\frac{385}{4096}a^{25}+\frac{1683}{4096}a^{24}-\frac{2289}{2048}a^{23}+\frac{8203}{4096}a^{22}-\frac{5357}{2048}a^{21}+\frac{5077}{2048}a^{20}-\frac{439}{512}a^{19}-\frac{7487}{2048}a^{18}+\frac{11303}{1024}a^{17}-\frac{10601}{512}a^{16}+\frac{15575}{512}a^{15}-\frac{36565}{1024}a^{14}+\frac{15229}{512}a^{13}-\frac{2165}{512}a^{12}-\frac{333}{8}a^{11}+\frac{1739}{16}a^{10}-\frac{12009}{64}a^{9}+\frac{14701}{64}a^{8}-\frac{413}{2}a^{7}+\frac{2101}{16}a^{6}+\frac{299}{8}a^{5}-310a^{4}+433a^{3}-465a^{2}+465a-325$, $\frac{29}{8192}a^{31}-\frac{101}{8192}a^{30}+\frac{41}{2048}a^{29}-\frac{41}{1024}a^{28}+\frac{123}{2048}a^{27}-\frac{81}{4096}a^{26}-\frac{117}{1024}a^{25}+\frac{593}{2048}a^{24}-\frac{179}{256}a^{23}+\frac{2435}{2048}a^{22}-\frac{1461}{1024}a^{21}+\frac{1125}{1024}a^{20}-\frac{3}{512}a^{19}-\frac{2915}{1024}a^{18}+\frac{929}{128}a^{17}-\frac{3259}{256}a^{16}+\frac{4545}{256}a^{15}-\frac{10005}{512}a^{14}+\frac{451}{32}a^{13}+\frac{821}{256}a^{12}-\frac{8027}{256}a^{11}+\frac{8691}{128}a^{10}-\frac{1817}{16}a^{9}+\frac{4227}{32}a^{8}-\frac{429}{4}a^{7}+\frac{487}{8}a^{6}+\frac{65}{4}a^{5}-\frac{795}{4}a^{4}+265a^{3}-265a^{2}+274a-251$, $\frac{15}{32768}a^{31}-\frac{53}{8192}a^{30}+\frac{161}{16384}a^{29}-\frac{11}{512}a^{28}+\frac{599}{16384}a^{27}-\frac{161}{4096}a^{26}-\frac{5}{128}a^{25}+\frac{453}{4096}a^{24}-\frac{2711}{8192}a^{23}+\frac{651}{1024}a^{22}-\frac{231}{256}a^{21}+\frac{845}{1024}a^{20}-\frac{1993}{4096}a^{19}-\frac{861}{1024}a^{18}+\frac{6401}{2048}a^{17}-\frac{799}{128}a^{16}+\frac{19623}{2048}a^{15}-\frac{189}{16}a^{14}+\frac{2763}{256}a^{13}-\frac{1941}{512}a^{12}-\frac{715}{64}a^{11}+\frac{3659}{128}a^{10}-\frac{1819}{32}a^{9}+\frac{587}{8}a^{8}-\frac{1095}{16}a^{7}+37a^{6}-\frac{43}{2}a^{5}-88a^{4}+132a^{3}-139a^{2}+118a-162$, $\frac{87}{32768}a^{31}-\frac{433}{16384}a^{30}+\frac{703}{16384}a^{29}-\frac{401}{4096}a^{28}+\frac{2751}{16384}a^{27}-\frac{1439}{8192}a^{26}-\frac{585}{4096}a^{25}+\frac{63}{128}a^{24}-\frac{12727}{8192}a^{23}+\frac{12351}{4096}a^{22}-\frac{8817}{2048}a^{21}+\frac{8423}{2048}a^{20}-\frac{10085}{4096}a^{19}-\frac{8257}{2048}a^{18}+\frac{31203}{2048}a^{17}-\frac{15683}{512}a^{16}+\frac{96687}{2048}a^{15}-\frac{59797}{1024}a^{14}+\frac{27507}{512}a^{13}-\frac{5045}{256}a^{12}-\frac{6809}{128}a^{11}+\frac{9391}{64}a^{10}-\frac{18351}{64}a^{9}+\frac{11955}{32}a^{8}-\frac{11495}{32}a^{7}+\frac{1731}{8}a^{6}-\frac{321}{4}a^{5}-\frac{3635}{8}a^{4}+\frac{2751}{4}a^{3}-758a^{2}+686a-778$, $\frac{99}{32768}a^{31}-\frac{207}{16384}a^{30}+\frac{445}{16384}a^{29}-\frac{59}{1024}a^{28}+\frac{1527}{16384}a^{27}-\frac{601}{8192}a^{26}-\frac{33}{512}a^{25}+\frac{173}{512}a^{24}-\frac{7627}{8192}a^{23}+\frac{6989}{4096}a^{22}-\frac{2343}{1024}a^{21}+\frac{4507}{2048}a^{20}-\frac{3561}{4096}a^{19}-\frac{5947}{2048}a^{18}+\frac{18729}{2048}a^{17}-\frac{557}{32}a^{16}+\frac{52935}{2048}a^{15}-\frac{31431}{1024}a^{14}+\frac{419}{16}a^{13}-\frac{1331}{256}a^{12}-\frac{2225}{64}a^{11}+\frac{5821}{64}a^{10}-\frac{5047}{32}a^{9}+\frac{12585}{64}a^{8}-\frac{709}{4}a^{7}+\frac{1649}{16}a^{6}+\frac{255}{8}a^{5}-\frac{2049}{8}a^{4}+\frac{737}{2}a^{3}-\frac{777}{2}a^{2}+364a-262$, $\frac{35}{16384}a^{31}-\frac{69}{16384}a^{30}+\frac{23}{4096}a^{29}-\frac{59}{4096}a^{28}+\frac{191}{8192}a^{27}+\frac{5}{8192}a^{26}-\frac{131}{4096}a^{25}+\frac{327}{4096}a^{24}-\frac{1019}{4096}a^{23}+\frac{1881}{4096}a^{22}-\frac{1143}{2048}a^{21}+\frac{1131}{2048}a^{20}-\frac{503}{2048}a^{19}-\frac{1705}{2048}a^{18}+\frac{335}{128}a^{17}-\frac{2555}{512}a^{16}+\frac{7599}{1024}a^{15}-\frac{9055}{1024}a^{14}+\frac{3883}{512}a^{13}-\frac{829}{512}a^{12}-\frac{65}{8}a^{11}+\frac{1611}{64}a^{10}-48a^{9}+\frac{3803}{64}a^{8}-\frac{451}{8}a^{7}+\frac{887}{16}a^{6}-\frac{41}{4}a^{5}-82a^{4}+\frac{455}{4}a^{3}-\frac{257}{2}a^{2}+176a-133$, $\frac{251}{32768}a^{31}-\frac{27}{1024}a^{30}+\frac{783}{16384}a^{29}-\frac{379}{4096}a^{28}+\frac{2223}{16384}a^{27}-\frac{23}{512}a^{26}-\frac{1035}{4096}a^{25}+\frac{2925}{4096}a^{24}-\frac{13499}{8192}a^{23}+\frac{5607}{2048}a^{22}-\frac{6617}{2048}a^{21}+\frac{2457}{1024}a^{20}+\frac{1827}{4096}a^{19}-\frac{913}{128}a^{18}+\frac{35343}{2048}a^{17}-\frac{15119}{512}a^{16}+\frac{82623}{2048}a^{15}-\frac{22131}{512}a^{14}+\frac{14863}{512}a^{13}+\frac{6399}{512}a^{12}-\frac{10093}{128}a^{11}+\frac{10501}{64}a^{10}-\frac{8325}{32}a^{9}+\frac{9463}{32}a^{8}-\frac{1827}{8}a^{7}+\frac{829}{8}a^{6}+\frac{393}{4}a^{5}-\frac{3699}{8}a^{4}+\frac{2415}{4}a^{3}-585a^{2}+562a-450$ (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: | \( 23735246685.523792 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 23735246685.523792 \cdot 5}{40\cdot\sqrt{7669926418924454281216000000000000000000000000}}\cr\approx \mathstrut & 0.199887693977467 \end{aligned}\] (assuming GRH)
Galois group
$C_4^2:C_2^2$ (as 32T204):
A solvable group of order 64 |
The 40 conjugacy class representatives for $C_4^2:C_2^2$ |
Character table for $C_4^2:C_2^2$ is not computed |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 32 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{8}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{16}$ | ${\href{/padicField/23.4.0.1}{4} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{16}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{16}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{16}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | Deg $16$ | $4$ | $4$ | $32$ | |||
Deg $16$ | $4$ | $4$ | $32$ | ||||
\(5\) | 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 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ | |
\(17\) | 17.4.0.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
17.4.0.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
17.4.0.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
17.4.0.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
17.8.4.1 | $x^{8} + 612 x^{7} + 140536 x^{6} + 14363966 x^{5} + 553913435 x^{4} + 345855654 x^{3} + 4032327212 x^{2} + 6379401496 x + 2294776272$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
17.8.4.1 | $x^{8} + 612 x^{7} + 140536 x^{6} + 14363966 x^{5} + 553913435 x^{4} + 345855654 x^{3} + 4032327212 x^{2} + 6379401496 x + 2294776272$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |