Normalized defining polynomial
\( x^{20} + x^{14} - 17x^{12} + 3x^{10} + 34x^{8} - 41x^{6} + 25x^{4} - 8x^{2} + 1 \)
Invariants
Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[8, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(12164157316311040000000000\) \(\medspace = 2^{20}\cdot 5^{10}\cdot 1039^{2}\cdot 1049^{2}\) | 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: | \(17.96\) | 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: | \(2\), \(5\), \(1039\), \(1049\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{1369}a^{18}-\frac{416}{1369}a^{16}+\frac{562}{1369}a^{14}+\frac{308}{1369}a^{12}+\frac{541}{1369}a^{10}-\frac{537}{1369}a^{8}+\frac{279}{1369}a^{6}+\frac{260}{1369}a^{4}+\frac{16}{1369}a^{2}+\frac{181}{1369}$, $\frac{1}{1369}a^{19}-\frac{416}{1369}a^{17}+\frac{562}{1369}a^{15}+\frac{308}{1369}a^{13}+\frac{541}{1369}a^{11}-\frac{537}{1369}a^{9}+\frac{279}{1369}a^{7}+\frac{260}{1369}a^{5}+\frac{16}{1369}a^{3}+\frac{181}{1369}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $13$ | 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: | $\frac{48773}{1369}a^{18}+\frac{16809}{1369}a^{16}+\frac{5784}{1369}a^{14}+\frac{50700}{1369}a^{12}-\frac{811730}{1369}a^{10}-\frac{133555}{1369}a^{8}+\frac{1612489}{1369}a^{6}-\frac{1442993}{1369}a^{4}+\frac{722870}{1369}a^{2}-\frac{141775}{1369}$, $a$, $\frac{170970}{1369}a^{19}+\frac{60373}{1369}a^{17}+\frac{21041}{1369}a^{15}+\frac{178145}{1369}a^{13}-\frac{2843759}{1369}a^{11}-\frac{491745}{1369}a^{9}+\frac{5643581}{1369}a^{7}-\frac{5013877}{1369}a^{5}+\frac{2495945}{1369}a^{3}-\frac{481194}{1369}a$, $\frac{110597}{1369}a^{19}+\frac{39332}{1369}a^{17}+\frac{13866}{1369}a^{15}+\frac{115414}{1369}a^{13}-\frac{1839104}{1369}a^{11}-\frac{322346}{1369}a^{9}+\frac{3647688}{1369}a^{7}-\frac{3235572}{1369}a^{5}+\frac{1609379}{1369}a^{3}-\frac{311593}{1369}a$, $\frac{128705}{1369}a^{18}+\frac{45487}{1369}a^{16}+\frac{16154}{1369}a^{14}+\frac{134538}{1369}a^{12}-\frac{2140420}{1369}a^{10}-\frac{370250}{1369}a^{8}+\frac{4243725}{1369}a^{6}-\frac{3778976}{1369}a^{4}+\frac{1884048}{1369}a^{2}-\frac{363453}{1369}$, $\frac{38778}{1369}a^{18}+\frac{12969}{1369}a^{16}+\frac{4232}{1369}a^{14}+\frac{40169}{1369}a^{12}-\frac{645826}{1369}a^{10}-\frac{99864}{1369}a^{8}+\frac{1286715}{1369}a^{6}-\frac{1159948}{1369}a^{4}+\frac{578009}{1369}a^{2}-\frac{112303}{1369}$, $a^{19}-\frac{147669}{1369}a^{18}-\frac{52855}{1369}a^{16}-\frac{18995}{1369}a^{14}+a^{13}-\frac{154462}{1369}a^{12}-17a^{11}+\frac{2455052}{1369}a^{10}+3a^{9}+\frac{435639}{1369}a^{8}+34a^{7}-\frac{4863653}{1369}a^{6}-41a^{5}+\frac{4313384}{1369}a^{4}+25a^{3}-\frac{2150509}{1369}a^{2}-8a+\frac{415074}{1369}$, $\frac{38778}{1369}a^{19}+\frac{12969}{1369}a^{17}+\frac{4232}{1369}a^{15}+\frac{40169}{1369}a^{13}-\frac{645826}{1369}a^{11}-\frac{99864}{1369}a^{9}+\frac{1286715}{1369}a^{7}-\frac{1159948}{1369}a^{5}+\frac{578009}{1369}a^{3}-\frac{112303}{1369}a+1$, $\frac{25224}{1369}a^{19}-\frac{3165}{37}a^{18}+\frac{8415}{1369}a^{17}-\frac{1115}{37}a^{16}+\frac{2631}{1369}a^{15}-\frac{399}{37}a^{14}+\frac{25928}{1369}a^{13}-\frac{3311}{37}a^{12}-\frac{420291}{1369}a^{11}+\frac{52635}{37}a^{10}-\frac{64745}{1369}a^{9}+\frac{9038}{37}a^{8}+\frac{838664}{1369}a^{7}-\frac{104333}{37}a^{6}-\frac{752220}{1369}a^{5}+\frac{93072}{37}a^{4}+\frac{374835}{1369}a^{3}-\frac{46533}{37}a^{2}-\frac{75366}{1369}a+\frac{9034}{37}$, $\frac{25224}{1369}a^{19}-\frac{48773}{1369}a^{18}+\frac{8415}{1369}a^{17}-\frac{16809}{1369}a^{16}+\frac{2631}{1369}a^{15}-\frac{5784}{1369}a^{14}+\frac{25928}{1369}a^{13}-\frac{50700}{1369}a^{12}-\frac{420291}{1369}a^{11}+\frac{811730}{1369}a^{10}-\frac{64745}{1369}a^{9}+\frac{133555}{1369}a^{8}+\frac{838664}{1369}a^{7}-\frac{1612489}{1369}a^{6}-\frac{752220}{1369}a^{5}+\frac{1442993}{1369}a^{4}+\frac{374835}{1369}a^{3}-\frac{722870}{1369}a^{2}-\frac{75366}{1369}a+\frac{141775}{1369}$, $\frac{25224}{1369}a^{19}+\frac{8415}{1369}a^{17}+\frac{2631}{1369}a^{15}+\frac{25928}{1369}a^{13}-\frac{420291}{1369}a^{11}-\frac{64745}{1369}a^{9}+\frac{838664}{1369}a^{7}-\frac{752220}{1369}a^{5}+\frac{374835}{1369}a^{3}-\frac{75366}{1369}a-1$, $\frac{170970}{1369}a^{19}+\frac{89002}{1369}a^{18}+\frac{60373}{1369}a^{17}+\frac{31260}{1369}a^{16}+\frac{21041}{1369}a^{15}+\frac{10923}{1369}a^{14}+\frac{178145}{1369}a^{13}+\frac{92852}{1369}a^{12}-\frac{2843759}{1369}a^{11}-\frac{1480275}{1369}a^{10}-\frac{491745}{1369}a^{9}-\frac{252811}{1369}a^{8}+\frac{5643581}{1369}a^{7}+\frac{2938510}{1369}a^{6}-\frac{5013877}{1369}a^{5}-\frac{2617215}{1369}a^{4}+\frac{2495945}{1369}a^{3}+\frac{1302191}{1369}a^{2}-\frac{482563}{1369}a-\frac{252926}{1369}$, $\frac{88448}{1369}a^{19}+\frac{48773}{1369}a^{18}+\frac{30363}{1369}a^{17}+\frac{16809}{1369}a^{16}+\frac{10338}{1369}a^{15}+\frac{5784}{1369}a^{14}+\frac{91976}{1369}a^{13}+\frac{50700}{1369}a^{12}-\frac{1471964}{1369}a^{11}-\frac{811730}{1369}a^{10}-\frac{240065}{1369}a^{9}-\frac{133555}{1369}a^{8}+\frac{2926320}{1369}a^{7}+\frac{1612489}{1369}a^{6}-\frac{2621617}{1369}a^{5}-\frac{1442993}{1369}a^{4}+\frac{1307017}{1369}a^{3}+\frac{722870}{1369}a^{2}-\frac{256001}{1369}a-\frac{141775}{1369}$ (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: | \( 72101.1365074 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{6}\cdot 72101.1365074 \cdot 1}{2\cdot\sqrt{12164157316311040000000000}}\cr\approx \mathstrut & 0.162813448911 \end{aligned}\] (assuming GRH)
Galois group
$C_2^8.S_5\wr C_2$ (as 20T1030):
A non-solvable group of order 7372800 |
The 189 conjugacy class representatives for $C_2^8.S_5\wr C_2$ |
Character table for $C_2^8.S_5\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 10.4.3405971875.1 |
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 32 sibling: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | 20.2.15749078053421278159366444959242240000000000.3 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | R | $16{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $20$ | $2$ | $10$ | $20$ | |||
\(5\) | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(1039\) | $\Q_{1039}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1039}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{1039}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{1039}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
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 $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
\(1049\) | $\Q_{1049}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1049}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
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 $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |