Normalized defining polynomial
\( x^{20} - 5 x^{19} + 5 x^{18} + 15 x^{17} - 30 x^{16} - 21 x^{15} + 75 x^{14} + 15 x^{13} - 120 x^{12} + \cdots + 1 \)
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: | \(781250000000000000000\) \(\medspace = 2^{16}\cdot 5^{23}\) | 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: | \(11.08\) | 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: | $2^{4/5}5^{23/20}\approx 11.08254495193135$ | ||
Ramified primes: | \(2\), \(5\) | 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: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Gal(K/\Q) }$: | $20$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{431}a^{18}-\frac{86}{431}a^{17}+\frac{74}{431}a^{16}+\frac{141}{431}a^{15}+\frac{112}{431}a^{14}-\frac{183}{431}a^{13}+\frac{132}{431}a^{12}-\frac{150}{431}a^{11}-\frac{170}{431}a^{10}+\frac{123}{431}a^{9}-\frac{170}{431}a^{8}-\frac{150}{431}a^{7}+\frac{132}{431}a^{6}-\frac{183}{431}a^{5}+\frac{112}{431}a^{4}+\frac{141}{431}a^{3}+\frac{74}{431}a^{2}-\frac{86}{431}a+\frac{1}{431}$, $\frac{1}{431}a^{19}+\frac{5}{431}a^{17}+\frac{40}{431}a^{16}+\frac{170}{431}a^{15}-\frac{33}{431}a^{14}-\frac{90}{431}a^{13}-\frac{4}{431}a^{12}-\frac{140}{431}a^{11}+\frac{157}{431}a^{10}+\frac{64}{431}a^{9}-\frac{116}{431}a^{8}+\frac{162}{431}a^{7}-\frac{37}{431}a^{6}-\frac{110}{431}a^{5}-\frac{140}{431}a^{4}+\frac{132}{431}a^{3}-\frac{187}{431}a^{2}-\frac{68}{431}a+\frac{86}{431}$
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: | \( \frac{288}{431} a^{19} + \frac{532}{431} a^{18} - \frac{6384}{431} a^{17} + \frac{5633}{431} a^{16} + \frac{23980}{431} a^{15} - \frac{30948}{431} a^{14} - \frac{54316}{431} a^{13} + \frac{77692}{431} a^{12} + \frac{93656}{431} a^{11} - \frac{121080}{431} a^{10} - \frac{133356}{431} a^{9} + \frac{124408}{431} a^{8} + \frac{140980}{431} a^{7} - \frac{80076}{431} a^{6} - \frac{92832}{431} a^{5} + \frac{43400}{431} a^{4} + \frac{38896}{431} a^{3} - \frac{17936}{431} a^{2} - \frac{8444}{431} a + \frac{4612}{431} \) (order $10$) | 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: | $\frac{2038}{431}a^{19}-\frac{7860}{431}a^{18}+\frac{429}{431}a^{17}+\frac{34320}{431}a^{16}-\frac{23497}{431}a^{15}-\frac{80833}{431}a^{14}+\frac{72296}{431}a^{13}+\frac{135699}{431}a^{12}-\frac{118739}{431}a^{11}-\frac{180325}{431}a^{10}+\frac{124351}{431}a^{9}+\frac{176159}{431}a^{8}-\frac{85113}{431}a^{7}-\frac{107834}{431}a^{6}+\frac{54810}{431}a^{5}+\frac{42022}{431}a^{4}-\frac{27240}{431}a^{3}-\frac{6788}{431}a^{2}+\frac{7246}{431}a-\frac{1113}{431}$, $\frac{2785}{431}a^{19}-\frac{12735}{431}a^{18}+\frac{8361}{431}a^{17}+\frac{45664}{431}a^{16}-\frac{63234}{431}a^{15}-\frac{88598}{431}a^{14}+\frac{170525}{431}a^{13}+\frac{124074}{431}a^{12}-\frac{282954}{431}a^{11}-\frac{153619}{431}a^{10}+\frac{331956}{431}a^{9}+\frac{154956}{431}a^{8}-\frac{272852}{431}a^{7}-\frac{103596}{431}a^{6}+\frac{166545}{431}a^{5}+\frac{32770}{431}a^{4}-\frac{70365}{431}a^{3}+\frac{2646}{431}a^{2}+\frac{16677}{431}a-\frac{4672}{431}$, $\frac{5469}{431}a^{19}-\frac{23941}{431}a^{18}+\frac{12299}{431}a^{17}+\frac{90098}{431}a^{16}-\frac{107345}{431}a^{15}-\frac{184497}{431}a^{14}+\frac{294453}{431}a^{13}+\frac{274532}{431}a^{12}-\frac{485883}{431}a^{11}-\frac{348561}{431}a^{10}+\frac{556745}{431}a^{9}+\frac{346158}{431}a^{8}-\frac{442737}{431}a^{7}-\frac{221437}{431}a^{6}+\frac{271704}{431}a^{5}+\frac{73356}{431}a^{4}-\frac{118631}{431}a^{3}-\frac{595}{431}a^{2}+\frac{28115}{431}a-\frac{7450}{431}$, $\frac{4675}{431}a^{19}-\frac{19961}{431}a^{18}+\frac{8263}{431}a^{17}+\frac{78311}{431}a^{16}-\frac{83699}{431}a^{15}-\frac{167240}{431}a^{14}+\frac{234508}{431}a^{13}+\frac{260008}{431}a^{12}-\frac{387287}{431}a^{11}-\frac{339108}{431}a^{10}+\frac{436462}{431}a^{9}+\frac{341788}{431}a^{8}-\frac{335673}{431}a^{7}-\frac{222258}{431}a^{6}+\frac{201348}{431}a^{5}+\frac{79889}{431}a^{4}-\frac{88949}{431}a^{3}-\frac{7992}{431}a^{2}+\frac{21270}{431}a-\frac{4518}{431}$, $\frac{3254}{431}a^{19}-\frac{16119}{431}a^{18}+\frac{14253}{431}a^{17}+\frac{54506}{431}a^{16}-\frac{96022}{431}a^{15}-\frac{93459}{431}a^{14}+\frac{255816}{431}a^{13}+\frac{107803}{431}a^{12}-\frac{431484}{431}a^{11}-\frac{115002}{431}a^{10}+\frac{526728}{431}a^{9}+\frac{114239}{431}a^{8}-\frac{460333}{431}a^{7}-\frac{78021}{431}a^{6}+\frac{289004}{431}a^{5}+\frac{11344}{431}a^{4}-\frac{124854}{431}a^{3}+\frac{16223}{431}a^{2}+\frac{29277}{431}a-\frac{9529}{431}$, $\frac{7450}{431}a^{19}-\frac{31781}{431}a^{18}+\frac{13309}{431}a^{17}+\frac{124049}{431}a^{16}-\frac{133402}{431}a^{15}-\frac{263795}{431}a^{14}+\frac{374253}{431}a^{13}+\frac{406203}{431}a^{12}-\frac{619468}{431}a^{11}-\frac{523133}{431}a^{10}+\frac{701889}{431}a^{9}+\frac{519495}{431}a^{8}-\frac{547842}{431}a^{7}-\frac{330987}{431}a^{6}+\frac{337313}{431}a^{5}+\frac{115254}{431}a^{4}-\frac{150144}{431}a^{3}-\frac{6881}{431}a^{2}+\frac{36655}{431}a-\frac{8704}{431}$, $\frac{7270}{431}a^{19}-\frac{30499}{431}a^{18}+\frac{11631}{431}a^{17}+\frac{119052}{431}a^{16}-\frac{122453}{431}a^{15}-\frac{253053}{431}a^{14}+\frac{342039}{431}a^{13}+\frac{389093}{431}a^{12}-\frac{559862}{431}a^{11}-\frac{496079}{431}a^{10}+\frac{625228}{431}a^{9}+\frac{479750}{431}a^{8}-\frac{482687}{431}a^{7}-\frac{291299}{431}a^{6}+\frac{304820}{431}a^{5}+\frac{99139}{431}a^{4}-\frac{135372}{431}a^{3}-\frac{3343}{431}a^{2}+\frac{32601}{431}a-\frac{8679}{431}$, $\frac{1319}{431}a^{19}-\frac{4403}{431}a^{18}-\frac{2647}{431}a^{17}+\frac{23466}{431}a^{16}-\frac{3952}{431}a^{15}-\frac{64287}{431}a^{14}+\frac{22868}{431}a^{13}+\frac{120368}{431}a^{12}-\frac{38824}{431}a^{11}-\frac{169317}{431}a^{10}+\frac{29446}{431}a^{9}+\frac{171402}{431}a^{8}-\frac{2095}{431}a^{7}-\frac{111075}{431}a^{6}-\frac{495}{431}a^{5}+\frac{49732}{431}a^{4}-\frac{3216}{431}a^{3}-\frac{13468}{431}a^{2}+\frac{1920}{431}a+\frac{850}{431}$, $\frac{1459}{431}a^{19}-\frac{7050}{431}a^{18}+\frac{5885}{431}a^{17}+\frac{23690}{431}a^{16}-\frac{39611}{431}a^{15}-\frac{41259}{431}a^{14}+\frac{104614}{431}a^{13}+\frac{48830}{431}a^{12}-\frac{173833}{431}a^{11}-\frac{52060}{431}a^{10}+\frac{208044}{431}a^{9}+\frac{47869}{431}a^{8}-\frac{177576}{431}a^{7}-\frac{27332}{431}a^{6}+\frac{111638}{431}a^{5}-\frac{836}{431}a^{4}-\frac{48936}{431}a^{3}+\frac{9282}{431}a^{2}+\frac{11438}{431}a-\frac{4411}{431}$ | 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: | \( 525.931286845 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 525.931286845 \cdot 1}{10\cdot\sqrt{781250000000000000000}}\cr\approx \mathstrut & 0.180439940617 \end{aligned}\]
Galois group
A solvable group of order 20 |
The 5 conjugacy class representatives for $F_5$ |
Character table for $F_5$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.50000.1 x5, 10.2.12500000000.1 x5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.1.50000.1 |
Degree 10 sibling: | 10.2.12500000000.1 |
Minimal sibling: | 5.1.50000.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{5}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}$ | ${\href{/padicField/11.5.0.1}{5} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{5}$ | ${\href{/padicField/17.4.0.1}{4} }^{5}$ | ${\href{/padicField/19.2.0.1}{2} }^{10}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.2.0.1}{2} }^{10}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | ${\href{/padicField/47.4.0.1}{4} }^{5}$ | ${\href{/padicField/53.4.0.1}{4} }^{5}$ | ${\href{/padicField/59.2.0.1}{2} }^{10}$ |
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 $20$ | $5$ | $4$ | $16$ | |||
\(5\) | Deg $20$ | $20$ | $1$ | $23$ |