Normalized defining polynomial
\( x^{20} - 3 x^{19} + 10 x^{18} - 26 x^{17} + 59 x^{16} - 129 x^{15} + 241 x^{14} - 438 x^{13} + \cdots + 1024 \)
Invariants
Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(22762830184956456665916278633281\) \(\medspace = 8311^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(36.97\) | 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: | $8311^{1/2}\approx 91.16468614545876$ | ||
Ramified primes: | \(8311\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | unavailable$^{512}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{2}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{12}-\frac{1}{4}a^{10}-\frac{1}{8}a^{9}-\frac{3}{8}a^{8}+\frac{3}{8}a^{7}+\frac{1}{8}a^{5}-\frac{1}{4}a^{4}+\frac{1}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{16}a^{14}-\frac{1}{16}a^{13}-\frac{1}{8}a^{11}+\frac{7}{16}a^{10}-\frac{3}{16}a^{9}+\frac{3}{16}a^{8}-\frac{1}{2}a^{7}+\frac{1}{16}a^{6}-\frac{1}{8}a^{5}-\frac{7}{16}a^{4}-\frac{3}{8}a^{3}$, $\frac{1}{32}a^{15}-\frac{1}{32}a^{14}-\frac{1}{16}a^{12}+\frac{7}{32}a^{11}-\frac{3}{32}a^{10}-\frac{13}{32}a^{9}-\frac{1}{4}a^{8}+\frac{1}{32}a^{7}+\frac{7}{16}a^{6}-\frac{7}{32}a^{5}+\frac{5}{16}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{16}-\frac{1}{64}a^{15}-\frac{1}{32}a^{13}+\frac{7}{64}a^{12}-\frac{3}{64}a^{11}+\frac{19}{64}a^{10}-\frac{1}{8}a^{9}-\frac{31}{64}a^{8}-\frac{9}{32}a^{7}-\frac{7}{64}a^{6}+\frac{5}{32}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{128}a^{17}-\frac{1}{128}a^{16}-\frac{1}{64}a^{14}+\frac{7}{128}a^{13}-\frac{3}{128}a^{12}+\frac{19}{128}a^{11}-\frac{1}{16}a^{10}+\frac{33}{128}a^{9}-\frac{9}{64}a^{8}-\frac{7}{128}a^{7}-\frac{27}{64}a^{6}+\frac{3}{8}a^{5}-\frac{1}{8}a^{4}+\frac{3}{8}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{256}a^{18}-\frac{1}{256}a^{17}-\frac{1}{128}a^{15}+\frac{7}{256}a^{14}-\frac{3}{256}a^{13}+\frac{19}{256}a^{12}-\frac{1}{32}a^{11}+\frac{33}{256}a^{10}-\frac{9}{128}a^{9}-\frac{7}{256}a^{8}-\frac{27}{128}a^{7}-\frac{5}{16}a^{6}-\frac{1}{16}a^{5}+\frac{3}{16}a^{4}-\frac{1}{4}a^{3}$, $\frac{1}{49664}a^{19}-\frac{17}{49664}a^{18}-\frac{35}{12416}a^{17}-\frac{3}{24832}a^{16}-\frac{633}{49664}a^{15}+\frac{197}{49664}a^{14}-\frac{577}{49664}a^{13}-\frac{321}{12416}a^{12}+\frac{9005}{49664}a^{11}+\frac{1199}{24832}a^{10}-\frac{10563}{49664}a^{9}+\frac{6857}{24832}a^{8}-\frac{5851}{12416}a^{7}-\frac{609}{6208}a^{6}-\frac{637}{3104}a^{5}+\frac{31}{776}a^{4}+\frac{55}{388}a^{3}+\frac{87}{194}a^{2}-\frac{22}{97}a+\frac{14}{97}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{4}$, which has order $4$ (assuming GRH)
Unit group
Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -1 \) (order $2$) | 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: | $\frac{1231}{49664}a^{19}-\frac{5019}{49664}a^{18}+\frac{1689}{6208}a^{17}-\frac{16109}{24832}a^{16}+\frac{67393}{49664}a^{15}-\frac{127257}{49664}a^{14}+\frac{216645}{49664}a^{13}-\frac{20921}{3104}a^{12}+\frac{484995}{49664}a^{11}-\frac{316397}{24832}a^{10}+\frac{753099}{49664}a^{9}-\frac{411447}{24832}a^{8}+\frac{192893}{12416}a^{7}-\frac{76111}{6208}a^{6}+\frac{20953}{3104}a^{5}-\frac{1569}{1552}a^{4}-\frac{3591}{776}a^{3}+\frac{3801}{388}a^{2}-\frac{1881}{194}a+\frac{356}{97}$, $\frac{1}{512}a^{19}-\frac{3}{512}a^{18}+\frac{5}{256}a^{17}-\frac{13}{256}a^{16}+\frac{59}{512}a^{15}-\frac{129}{512}a^{14}+\frac{241}{512}a^{13}-\frac{219}{256}a^{12}+\frac{729}{512}a^{11}-\frac{281}{128}a^{10}+\frac{1669}{512}a^{9}-\frac{281}{64}a^{8}+\frac{729}{128}a^{7}-\frac{219}{32}a^{6}+\frac{241}{32}a^{5}-\frac{129}{16}a^{4}+\frac{59}{8}a^{3}-\frac{13}{2}a^{2}+4a-2$, $\frac{3}{512}a^{19}-\frac{7}{512}a^{18}+\frac{3}{64}a^{17}-\frac{29}{256}a^{16}+\frac{125}{512}a^{15}-\frac{269}{512}a^{14}+\frac{465}{512}a^{13}-\frac{13}{8}a^{12}+\frac{1311}{512}a^{11}-\frac{957}{256}a^{10}+\frac{2759}{512}a^{9}-\frac{1703}{256}a^{8}+\frac{1063}{128}a^{7}-\frac{585}{64}a^{6}+\frac{285}{32}a^{5}-\frac{73}{8}a^{4}+6a^{3}-\frac{23}{4}a^{2}+2a-1$, $\frac{15}{6208}a^{19}+\frac{435}{24832}a^{18}-\frac{349}{24832}a^{17}+\frac{887}{12416}a^{16}-\frac{657}{12416}a^{15}+\frac{1441}{24832}a^{14}+\frac{1173}{24832}a^{13}-\frac{15057}{24832}a^{12}+\frac{16689}{12416}a^{11}-\frac{83585}{24832}a^{10}+\frac{9495}{1552}a^{9}-\frac{250077}{24832}a^{8}+\frac{49995}{3104}a^{7}-\frac{129049}{6208}a^{6}+\frac{43357}{1552}a^{5}-\frac{47947}{1552}a^{4}+\frac{5919}{194}a^{3}-\frac{3025}{97}a^{2}+\frac{3741}{194}a-\frac{1715}{97}$, $\frac{39}{24832}a^{19}+\frac{1}{1552}a^{18}+\frac{69}{24832}a^{17}-\frac{505}{12416}a^{16}+\frac{2667}{24832}a^{15}-\frac{1935}{6208}a^{14}+\frac{4729}{6208}a^{13}-\frac{36399}{24832}a^{12}+\frac{74939}{24832}a^{11}-\frac{120751}{24832}a^{10}+\frac{195069}{24832}a^{9}-\frac{291691}{24832}a^{8}+\frac{185665}{12416}a^{7}-\frac{15565}{776}a^{6}+\frac{33357}{1552}a^{5}-\frac{18049}{776}a^{4}+\frac{17019}{776}a^{3}-\frac{1554}{97}a^{2}+\frac{1388}{97}a-\frac{460}{97}$, $\frac{879}{49664}a^{19}-\frac{3109}{49664}a^{18}+\frac{4527}{24832}a^{17}-\frac{10979}{24832}a^{16}+\frac{43829}{49664}a^{15}-\frac{84663}{49664}a^{14}+\frac{139031}{49664}a^{13}-\frac{107545}{24832}a^{12}+\frac{303223}{49664}a^{11}-\frac{91657}{12416}a^{10}+\frac{420555}{49664}a^{9}-\frac{12015}{1552}a^{8}+\frac{68199}{12416}a^{7}-\frac{2223}{1552}a^{6}-\frac{18081}{3104}a^{5}+\frac{3973}{388}a^{4}-\frac{12629}{776}a^{3}+\frac{7543}{388}a^{2}-\frac{1684}{97}a+\frac{1345}{97}$, $\frac{993}{49664}a^{19}-\frac{2719}{49664}a^{18}+\frac{3725}{24832}a^{17}-\frac{9187}{24832}a^{16}+\frac{34523}{49664}a^{15}-\frac{68413}{49664}a^{14}+\frac{110113}{49664}a^{13}-\frac{83927}{24832}a^{12}+\frac{247661}{49664}a^{11}-\frac{37847}{6208}a^{10}+\frac{375329}{49664}a^{9}-\frac{100449}{12416}a^{8}+\frac{11175}{1552}a^{7}-\frac{5061}{776}a^{6}+\frac{10181}{3104}a^{5}-\frac{1581}{1552}a^{4}-\frac{671}{776}a^{3}+\frac{1771}{388}a^{2}-\frac{527}{194}a+\frac{128}{97}$, $\frac{61}{49664}a^{19}-\frac{455}{49664}a^{18}+\frac{95}{24832}a^{17}+\frac{981}{24832}a^{16}-\frac{4857}{49664}a^{15}+\frac{19195}{49664}a^{14}-\frac{44703}{49664}a^{13}+\frac{49011}{24832}a^{12}-\frac{191775}{49664}a^{11}+\frac{78813}{12416}a^{10}-\frac{524451}{49664}a^{9}+\frac{93875}{6208}a^{8}-\frac{63159}{3104}a^{7}+\frac{81675}{3104}a^{6}-\frac{43921}{1552}a^{5}+\frac{47723}{1552}a^{4}-\frac{21517}{776}a^{3}+\frac{2120}{97}a^{2}-\frac{1827}{97}a+\frac{272}{97}$, $\frac{1}{256}a^{19}-\frac{11}{256}a^{18}+\frac{17}{128}a^{17}-\frac{49}{128}a^{16}+\frac{251}{256}a^{15}-\frac{513}{256}a^{14}+\frac{1049}{256}a^{13}-\frac{923}{128}a^{12}+\frac{3033}{256}a^{11}-\frac{1203}{64}a^{10}+\frac{6621}{256}a^{9}-\frac{1135}{32}a^{8}+\frac{2803}{64}a^{7}-\frac{1565}{32}a^{6}+\frac{1743}{32}a^{5}-\frac{395}{8}a^{4}+\frac{355}{8}a^{3}-\frac{143}{4}a^{2}+\frac{33}{2}a-16$ (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: | \( 10106391.1313 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 10106391.1313 \cdot 4}{2\cdot\sqrt{22762830184956456665916278633281}}\cr\approx \mathstrut & 0.406267098100 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times A_5$ (as 20T36):
A non-solvable group of order 120 |
The 10 conjugacy class representatives for $C_2\times A_5$ |
Character table for $C_2\times A_5$ |
Intermediate fields
10.10.4771040786343841.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 10 sibling: | data not computed |
Degree 12 siblings: | data not computed |
Degree 20 sibling: | data not computed |
Degree 24 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 40 sibling: | data not computed |
Minimal sibling: | 10.0.39652119975303662551.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{4}$ | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.5.0.1}{5} }^{4}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{10}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(8311\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ |