Normalized defining polynomial
\( x^{20} - 10x^{18} + 36x^{16} - 48x^{14} - 13x^{12} + 78x^{10} - 21x^{8} - 52x^{6} + 24x^{4} + 8x^{2} - 4 \)
Invariants
Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[10, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-1082081291410568717705150464\) \(\medspace = -\,2^{38}\cdot 89^{8}\) | 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: | \(22.48\) | 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))
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Galois root discriminant: | not computed | ||
Ramified primes: | \(2\), \(89\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{4}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{4}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{4}a^{9}-\frac{1}{2}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a$, $\frac{1}{8}a^{18}-\frac{1}{2}a^{15}-\frac{1}{4}a^{14}-\frac{1}{2}a^{13}-\frac{1}{8}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+\frac{1}{8}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{8}a^{19}-\frac{1}{4}a^{15}-\frac{1}{2}a^{14}-\frac{1}{8}a^{11}-\frac{1}{2}a^{9}+\frac{1}{8}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $14$ | 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)
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Fundamental units: | $\frac{3}{2}a^{18}-\frac{27}{2}a^{16}+41a^{14}-35a^{12}-\frac{89}{2}a^{10}+\frac{137}{2}a^{8}+\frac{47}{2}a^{6}-\frac{97}{2}a^{4}-a^{2}+8$, $\frac{3}{2}a^{19}-\frac{53}{4}a^{17}+\frac{77}{2}a^{15}-\frac{53}{2}a^{13}-\frac{107}{2}a^{11}+\frac{245}{4}a^{9}+38a^{7}-\frac{177}{4}a^{5}-10a^{3}+\frac{15}{2}a$, $a^{2}-1$, $\frac{7}{4}a^{18}-\frac{31}{2}a^{16}+\frac{91}{2}a^{14}-34a^{12}-\frac{223}{4}a^{10}+\frac{139}{2}a^{8}+\frac{135}{4}a^{6}-46a^{4}-\frac{9}{2}a^{2}+7$, $a^{19}-\frac{39}{4}a^{17}+\frac{67}{2}a^{15}-\frac{79}{2}a^{13}-21a^{11}+\frac{259}{4}a^{9}+\frac{5}{2}a^{7}-\frac{175}{4}a^{5}+4a^{3}+\frac{11}{2}a$, $\frac{3}{2}a^{19}-\frac{59}{4}a^{17}+\frac{103}{2}a^{15}-\frac{127}{2}a^{13}-\frac{55}{2}a^{11}+\frac{415}{4}a^{9}-8a^{7}-\frac{275}{4}a^{5}+14a^{3}+\frac{15}{2}a$, $\frac{3}{8}a^{18}-\frac{1}{4}a^{17}-4a^{16}+2a^{15}+\frac{61}{4}a^{14}-5a^{13}-21a^{12}+2a^{11}-\frac{59}{8}a^{10}+\frac{29}{4}a^{9}+\frac{71}{2}a^{8}-5a^{7}-\frac{13}{8}a^{6}-\frac{19}{4}a^{5}-\frac{101}{4}a^{4}+\frac{7}{2}a^{3}+\frac{9}{4}a^{2}+a+\frac{7}{2}$, $\frac{1}{4}a^{19}+\frac{5}{8}a^{18}-\frac{9}{4}a^{17}-\frac{23}{4}a^{16}+7a^{15}+\frac{73}{4}a^{14}-7a^{13}-\frac{35}{2}a^{12}-\frac{21}{4}a^{11}-\frac{157}{8}a^{10}+\frac{49}{4}a^{9}+\frac{153}{4}a^{8}-\frac{1}{4}a^{7}+\frac{65}{8}a^{6}-\frac{33}{4}a^{5}-\frac{59}{2}a^{4}+\frac{5}{2}a^{3}-\frac{1}{4}a^{2}+2a+6$, $\frac{5}{8}a^{18}+\frac{1}{2}a^{17}-\frac{21}{4}a^{16}-4a^{15}+\frac{57}{4}a^{14}+\frac{19}{2}a^{13}-\frac{17}{2}a^{12}-a^{11}-\frac{141}{8}a^{10}-\frac{37}{2}a^{9}+\frac{63}{4}a^{8}+6a^{7}+\frac{81}{8}a^{6}+15a^{5}-7a^{4}-2a^{3}-\frac{5}{4}a^{2}-\frac{5}{2}a$, $\frac{5}{8}a^{19}-\frac{1}{4}a^{18}-\frac{23}{4}a^{17}+\frac{5}{2}a^{16}+\frac{73}{4}a^{15}-9a^{14}-\frac{37}{2}a^{13}+12a^{12}-\frac{109}{8}a^{11}+\frac{13}{4}a^{10}+\frac{117}{4}a^{9}-\frac{39}{2}a^{8}+\frac{33}{8}a^{7}+\frac{21}{4}a^{6}-\frac{37}{2}a^{5}+12a^{4}+\frac{7}{4}a^{3}-4a^{2}+2a-1$, $\frac{5}{4}a^{19}-\frac{5}{8}a^{18}-\frac{45}{4}a^{17}+\frac{25}{4}a^{16}+34a^{15}-\frac{89}{4}a^{14}-28a^{13}+\frac{55}{2}a^{12}-\frac{157}{4}a^{11}+\frac{125}{8}a^{10}+\frac{225}{4}a^{9}-\frac{207}{4}a^{8}+\frac{91}{4}a^{7}-\frac{17}{8}a^{6}-\frac{149}{4}a^{5}+40a^{4}-\frac{7}{2}a^{3}-\frac{3}{4}a^{2}+5a-7$, $\frac{7}{8}a^{19}-\frac{7}{8}a^{18}-8a^{17}+\frac{15}{2}a^{16}+\frac{99}{4}a^{15}-\frac{83}{4}a^{14}-\frac{43}{2}a^{13}+12a^{12}-\frac{231}{8}a^{11}+\frac{239}{8}a^{10}+\frac{93}{2}a^{9}-28a^{8}+\frac{107}{8}a^{7}-\frac{155}{8}a^{6}-\frac{127}{4}a^{5}+\frac{63}{4}a^{4}+\frac{3}{4}a^{3}+\frac{17}{4}a^{2}+4a-\frac{5}{2}$, $\frac{5}{4}a^{19}-\frac{1}{4}a^{18}-\frac{23}{2}a^{17}+\frac{7}{4}a^{16}+36a^{15}-3a^{14}-33a^{13}-\frac{5}{2}a^{12}-\frac{149}{4}a^{11}+\frac{25}{4}a^{10}+\frac{127}{2}a^{9}+\frac{25}{4}a^{8}+\frac{71}{4}a^{7}-\frac{23}{4}a^{6}-41a^{5}-\frac{27}{4}a^{4}-2a^{3}+\frac{1}{2}a^{2}+5a+\frac{1}{2}$, $\frac{9}{8}a^{19}-\frac{43}{4}a^{17}-\frac{1}{4}a^{16}+\frac{143}{4}a^{15}+2a^{14}-\frac{77}{2}a^{13}-\frac{9}{2}a^{12}-\frac{241}{8}a^{11}-a^{10}+\frac{289}{4}a^{9}+\frac{45}{4}a^{8}+\frac{65}{8}a^{7}-a^{6}-\frac{107}{2}a^{5}-\frac{45}{4}a^{4}+\frac{13}{4}a^{3}+\frac{1}{2}a^{2}+10a+\frac{7}{2}$ (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)]
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Regulator: | \( 2261786.67414 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{5}\cdot 2261786.67414 \cdot 1}{2\cdot\sqrt{1082081291410568717705150464}}\cr\approx \mathstrut & 0.344739462024 \end{aligned}\] (assuming GRH)
Galois group
$C_2^9.C_2^4.S_5$ (as 20T965):
A non-solvable group of order 983040 |
The 149 conjugacy class representatives for $C_2^9.C_2^4.S_5$ |
Character table for $C_2^9.C_2^4.S_5$ |
Intermediate fields
5.3.31684.1, 10.6.256992219136.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 20 siblings: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | 20.0.270520322852642179426287616.3 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{4}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.14.3 | $x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{4} + 2$ | $8$ | $1$ | $14$ | $C_2^3 : C_4 $ | $[2, 2, 2]^{4}$ |
2.12.24.35 | $x^{12} + 2 x^{10} + 12 x^{9} + 78 x^{8} + 16 x^{7} + 312 x^{6} + 336 x^{5} + 748 x^{4} + 1472 x^{3} + 2872 x^{2} + 1264 x + 3624$ | $4$ | $3$ | $24$ | 12T141 | $[2, 2, 2, 3, 3, 3]^{6}$ | |
\(89\) | 89.8.0.1 | $x^{8} + 65 x^{3} + 40 x^{2} + 79 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |
89.12.8.1 | $x^{12} + 12 x^{10} + 572 x^{9} + 57 x^{8} + 1728 x^{7} - 52130 x^{6} + 480 x^{5} - 145881 x^{4} - 4313292 x^{3} + 1562790 x^{2} - 18240564 x + 371413136$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ |