Normalized defining polynomial
\( x^{20} - 3 x^{19} + 5 x^{18} - 5 x^{17} + 4 x^{16} - 5 x^{15} + 11 x^{14} - 21 x^{13} + 30 x^{12} + \cdots + 1 \)
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: | \(8200461879035261943808\) \(\medspace = 2^{16}\cdot 277^{7}\) | 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: | \(12.46\) | 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: | $2^{4/5}277^{1/2}\approx 28.97769793904893$ | ||
Ramified primes: | \(2\), \(277\) | 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: | \(\Q(\sqrt{277}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}$
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: | $a$, $a^{17}-\frac{3}{2}a^{16}+a^{15}+a^{13}-3a^{12}+\frac{11}{2}a^{11}-7a^{10}+7a^{9}-7a^{8}+9a^{7}-\frac{11}{2}a^{6}+3a^{5}-a^{4}-a^{2}+\frac{3}{2}a$, $\frac{3}{2}a^{19}-4a^{18}+5a^{17}-3a^{16}+2a^{15}-\frac{11}{2}a^{14}+14a^{13}-23a^{12}+28a^{11}-28a^{10}+\frac{59}{2}a^{9}-27a^{8}+20a^{7}-11a^{6}+4a^{5}-\frac{5}{2}a^{4}+6a^{3}-5a^{2}+3a$, $\frac{3}{2}a^{18}-\frac{7}{2}a^{17}+5a^{16}-3a^{15}+\frac{3}{2}a^{14}-\frac{9}{2}a^{13}+13a^{12}-22a^{11}+\frac{51}{2}a^{10}-\frac{53}{2}a^{9}+26a^{8}-23a^{7}+\frac{35}{2}a^{6}-\frac{13}{2}a^{5}+a^{4}-a^{3}+\frac{5}{2}a^{2}-\frac{9}{2}a+2$, $\frac{5}{2}a^{19}-\frac{11}{2}a^{18}+\frac{17}{2}a^{17}-7a^{16}+\frac{11}{2}a^{15}-8a^{14}+\frac{41}{2}a^{13}-38a^{12}+49a^{11}-54a^{10}+54a^{9}-48a^{8}+41a^{7}-\frac{49}{2}a^{6}+10a^{5}-\frac{9}{2}a^{4}+6a^{3}-\frac{17}{2}a^{2}+\frac{15}{2}a-\frac{7}{2}$, $\frac{5}{2}a^{19}-\frac{9}{2}a^{18}+4a^{17}-a^{16}+\frac{5}{2}a^{15}-8a^{14}+16a^{13}-\frac{45}{2}a^{12}+\frac{49}{2}a^{11}-25a^{10}+\frac{55}{2}a^{9}-\frac{41}{2}a^{8}+13a^{7}-6a^{6}+\frac{3}{2}a^{5}-3a^{4}+5a^{3}-\frac{5}{2}a^{2}+\frac{3}{2}a-1$, $\frac{5}{2}a^{19}-\frac{9}{2}a^{18}+4a^{17}-a^{16}+\frac{5}{2}a^{15}-8a^{14}+16a^{13}-\frac{45}{2}a^{12}+\frac{49}{2}a^{11}-25a^{10}+\frac{55}{2}a^{9}-\frac{41}{2}a^{8}+13a^{7}-6a^{6}+\frac{3}{2}a^{5}-3a^{4}+5a^{3}-\frac{5}{2}a^{2}+\frac{3}{2}a$, $a^{19}-2a^{18}+2a^{17}-a^{16}+\frac{1}{2}a^{15}-\frac{7}{2}a^{14}+7a^{13}-\frac{23}{2}a^{12}+11a^{11}-12a^{10}+\frac{25}{2}a^{9}-\frac{21}{2}a^{8}+\frac{15}{2}a^{7}-\frac{7}{2}a^{6}+a^{5}-3a^{4}+\frac{3}{2}a^{3}-3a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{3}{2}a^{19}-3a^{18}+3a^{17}+\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-4a^{14}+\frac{23}{2}a^{13}-\frac{29}{2}a^{12}+13a^{11}-10a^{10}+\frac{21}{2}a^{9}-8a^{8}+3a^{7}+\frac{9}{2}a^{6}-\frac{5}{2}a^{5}+\frac{7}{2}a^{3}-\frac{5}{2}a^{2}-a+2$ | 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: | \( 351.863063149 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 351.863063149 \cdot 1}{2\cdot\sqrt{8200461879035261943808}}\cr\approx \mathstrut & 0.186304442104 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
5.1.4432.1 x2, 10.2.5441006848.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.1.4432.1 |
Degree 6 sibling: | 6.2.340062928.2 |
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: | 5.1.4432.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{5}$ | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{5}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{5}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | ${\href{/padicField/47.2.0.1}{2} }^{10}$ | ${\href{/padicField/53.4.0.1}{4} }^{5}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 $20$ | $5$ | $4$ | $16$ | |||
\(277\) | 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 $4$ | $2$ | $2$ | $2$ |