Normalized defining polynomial
\( x^{20} - 5 x^{19} + 12 x^{18} - 21 x^{17} + 36 x^{16} - 63 x^{15} + 103 x^{14} - 143 x^{13} + 166 x^{12} + \cdots + 1 \)
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: | \(27918514644629391162169\) \(\medspace = 1609^{7}\) | 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: | \(13.25\) | 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: | $1609^{1/2}\approx 40.11234224026316$ | ||
Ramified primes: | \(1609\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{1609}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{73}a^{18}-\frac{27}{73}a^{17}+\frac{21}{73}a^{16}-\frac{18}{73}a^{15}-\frac{27}{73}a^{14}-\frac{35}{73}a^{13}+\frac{24}{73}a^{12}+\frac{21}{73}a^{11}-\frac{28}{73}a^{10}-\frac{13}{73}a^{9}-\frac{28}{73}a^{8}+\frac{21}{73}a^{7}+\frac{24}{73}a^{6}-\frac{35}{73}a^{5}-\frac{27}{73}a^{4}-\frac{18}{73}a^{3}+\frac{21}{73}a^{2}-\frac{27}{73}a+\frac{1}{73}$, $\frac{1}{73}a^{19}+\frac{22}{73}a^{17}-\frac{35}{73}a^{16}-\frac{2}{73}a^{15}-\frac{34}{73}a^{14}+\frac{28}{73}a^{13}+\frac{12}{73}a^{12}+\frac{28}{73}a^{11}+\frac{34}{73}a^{10}-\frac{14}{73}a^{9}-\frac{5}{73}a^{8}+\frac{7}{73}a^{7}+\frac{29}{73}a^{6}-\frac{23}{73}a^{5}-\frac{17}{73}a^{4}-\frac{27}{73}a^{3}+\frac{29}{73}a^{2}+\frac{2}{73}a+\frac{27}{73}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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)
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Fundamental units: | $\frac{130}{73}a^{19}-\frac{416}{73}a^{18}+\frac{660}{73}a^{17}-15a^{16}+\frac{2191}{73}a^{15}-\frac{3627}{73}a^{14}+\frac{5352}{73}a^{13}-\frac{6380}{73}a^{12}+\frac{6949}{73}a^{11}-\frac{7292}{73}a^{10}+\frac{7676}{73}a^{9}-\frac{7325}{73}a^{8}+\frac{6628}{73}a^{7}-\frac{5192}{73}a^{6}+\frac{3759}{73}a^{5}-\frac{2439}{73}a^{4}+\frac{1423}{73}a^{3}-\frac{732}{73}a^{2}+\frac{396}{73}a-\frac{191}{73}$, $\frac{234}{73}a^{19}-\frac{900}{73}a^{18}+\frac{1635}{73}a^{17}-\frac{2489}{73}a^{16}+\frac{4563}{73}a^{15}-\frac{8038}{73}a^{14}+\frac{12210}{73}a^{13}-\frac{14631}{73}a^{12}+\frac{14808}{73}a^{11}-\frac{14294}{73}a^{10}+\frac{14775}{73}a^{9}-\frac{14952}{73}a^{8}+\frac{13398}{73}a^{7}-\frac{9631}{73}a^{6}+\frac{5751}{73}a^{5}-\frac{3257}{73}a^{4}+\frac{1998}{73}a^{3}-\frac{1164}{73}a^{2}+\frac{459}{73}a-\frac{130}{73}$, $\frac{143}{73}a^{19}-\frac{680}{73}a^{18}+\frac{1504}{73}a^{17}-\frac{2495}{73}a^{16}+\frac{4289}{73}a^{15}-\frac{7526}{73}a^{14}+\frac{12109}{73}a^{13}-\frac{16210}{73}a^{12}+\frac{17756}{73}a^{11}-\frac{17562}{73}a^{10}+\frac{17277}{73}a^{9}-\frac{17591}{73}a^{8}+\frac{16870}{73}a^{7}-\frac{13852}{73}a^{6}+\frac{9050}{73}a^{5}-\frac{5168}{73}a^{4}+\frac{2831}{73}a^{3}-\frac{1811}{73}a^{2}+\frac{980}{73}a-\frac{323}{73}$, $\frac{288}{73}a^{19}-\frac{1308}{73}a^{18}+\frac{2816}{73}a^{17}-\frac{4625}{73}a^{16}+\frac{8003}{73}a^{15}-\frac{14042}{73}a^{14}+\frac{22454}{73}a^{13}-\frac{29615}{73}a^{12}+\frac{32207}{73}a^{11}-\frac{31694}{73}a^{10}+\frac{31368}{73}a^{9}-\frac{31830}{73}a^{8}+\frac{30539}{73}a^{7}-\frac{24500}{73}a^{6}+\frac{15869}{73}a^{5}-\frac{8927}{73}a^{4}+68a^{3}-\frac{3056}{73}a^{2}+\frac{1655}{73}a-\frac{467}{73}$, $\frac{17}{73}a^{19}-\frac{37}{73}a^{18}-\frac{87}{73}a^{17}+\frac{307}{73}a^{16}-\frac{390}{73}a^{15}+\frac{567}{73}a^{14}-\frac{1295}{73}a^{13}+\frac{2528}{73}a^{12}-\frac{3659}{73}a^{11}+\frac{3585}{73}a^{10}-\frac{3334}{73}a^{9}+\frac{3214}{73}a^{8}-\frac{3797}{73}a^{7}+\frac{3766}{73}a^{6}-\frac{2892}{73}a^{5}+\frac{1513}{73}a^{4}-\frac{888}{73}a^{3}+\frac{446}{73}a^{2}-\frac{500}{73}a+\frac{130}{73}$, $\frac{619}{73}a^{19}-\frac{2687}{73}a^{18}+\frac{5429}{73}a^{17}-\frac{8450}{73}a^{16}+\frac{14862}{73}a^{15}-\frac{26534}{73}a^{14}+\frac{41516}{73}a^{13}-\frac{52388}{73}a^{12}+\frac{54856}{73}a^{11}-\frac{53222}{73}a^{10}+\frac{53932}{73}a^{9}-\frac{54952}{73}a^{8}+\frac{51055}{73}a^{7}-\frac{38872}{73}a^{6}+\frac{24109}{73}a^{5}-\frac{13529}{73}a^{4}+\frac{8074}{73}a^{3}-\frac{4750}{73}a^{2}+\frac{2247}{73}a-\frac{501}{73}$, $\frac{348}{73}a^{19}-\frac{1569}{73}a^{18}+\frac{3299}{73}a^{17}-\frac{5271}{73}a^{16}+\frac{9223}{73}a^{15}-\frac{16408}{73}a^{14}+\frac{25896}{73}a^{13}-\frac{33553}{73}a^{12}+\frac{36217}{73}a^{11}-\frac{35705}{73}a^{10}+\frac{35819}{73}a^{9}-\frac{36210}{73}a^{8}+\frac{34165}{73}a^{7}-\frac{27272}{73}a^{6}+\frac{17784}{73}a^{5}-\frac{10127}{73}a^{4}+\frac{5779}{73}a^{3}-\frac{3439}{73}a^{2}+\frac{1814}{73}a-\frac{568}{73}$, $\frac{207}{73}a^{19}-\frac{800}{73}a^{18}+\frac{1480}{73}a^{17}-\frac{2218}{73}a^{16}+\frac{3912}{73}a^{15}-\frac{6973}{73}a^{14}+\frac{10801}{73}a^{13}-\frac{12628}{73}a^{12}+\frac{12210}{73}a^{11}-\frac{11369}{73}a^{10}+\frac{11809}{73}a^{9}-\frac{12288}{73}a^{8}+\frac{10929}{73}a^{7}-\frac{7065}{73}a^{6}+\frac{3748}{73}a^{5}-\frac{1921}{73}a^{4}+\frac{1292}{73}a^{3}-\frac{796}{73}a^{2}+\frac{187}{73}a+\frac{117}{73}$, $\frac{478}{73}a^{19}-\frac{1697}{73}a^{18}+\frac{2972}{73}a^{17}-\frac{4698}{73}a^{16}+\frac{8712}{73}a^{15}-\frac{14890}{73}a^{14}+\frac{22628}{73}a^{13}-\frac{26962}{73}a^{12}+\frac{27606}{73}a^{11}-\frac{26752}{73}a^{10}+\frac{27341}{73}a^{9}-\frac{27217}{73}a^{8}+\frac{24795}{73}a^{7}-\frac{17668}{73}a^{6}+\frac{10733}{73}a^{5}-\frac{5742}{73}a^{4}+\frac{3405}{73}a^{3}-\frac{2065}{73}a^{2}+\frac{931}{73}a-\frac{179}{73}$ | 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: | \( 751.110855868 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 751.110855868 \cdot 1}{2\cdot\sqrt{27918514644629391162169}}\cr\approx \mathstrut & 0.215539351819 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
5.1.1609.1 x2, 10.2.4165509529.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.1.1609.1 |
Degree 6 sibling: | 6.2.4165509529.1 |
Degree 10 siblings: | data not computed |
Degree 12 sibling: | data not computed |
Degree 15 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 40 sibling: | data not computed |
Minimal sibling: | 5.1.1609.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{4}$ | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.5.0.1}{5} }^{4}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{5}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.4.0.1}{4} }^{5}$ | ${\href{/padicField/31.2.0.1}{2} }^{10}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | ${\href{/padicField/47.4.0.1}{4} }^{5}$ | ${\href{/padicField/53.4.0.1}{4} }^{5}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(1609\) | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ |