Normalized defining polynomial
\( x^{10} - 2x^{9} - 7x^{8} + 12x^{7} + 21x^{6} - 14x^{5} - 47x^{4} - 24x^{3} + 56x^{2} + 64x + 16 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-161960834367488\) \(\medspace = -\,2^{25}\cdot 13^{6}\) | 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: | \(26.36\) | 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: | $2^{11/4}13^{2/3}\approx 37.1930153725407$ | ||
Ramified primes: | \(2\), \(13\) | 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: | \(\Q(\sqrt{-2}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{16}a^{8}+\frac{1}{8}a^{7}-\frac{3}{16}a^{6}-\frac{7}{16}a^{4}+\frac{3}{8}a^{3}+\frac{5}{16}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{64}a^{9}-\frac{15}{64}a^{7}-\frac{1}{32}a^{6}+\frac{9}{64}a^{5}+\frac{1}{16}a^{4}+\frac{17}{64}a^{3}-\frac{3}{32}a^{2}-\frac{7}{16}a-\frac{3}{8}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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: | $\frac{3}{64}a^{9}-\frac{3}{16}a^{8}-\frac{5}{64}a^{7}+\frac{31}{32}a^{6}-\frac{5}{64}a^{5}-\frac{3}{2}a^{4}-\frac{85}{64}a^{3}+\frac{41}{32}a^{2}+\frac{63}{16}a+\frac{5}{8}$, $\frac{7}{64}a^{9}-\frac{1}{4}a^{8}-\frac{41}{64}a^{7}+\frac{49}{32}a^{6}+\frac{95}{64}a^{5}-\frac{45}{16}a^{4}-\frac{201}{64}a^{3}+\frac{67}{32}a^{2}+\frac{71}{16}a+\frac{11}{8}$, $\frac{3}{8}a^{9}-\frac{17}{16}a^{8}-\frac{7}{4}a^{7}+\frac{95}{16}a^{6}+\frac{23}{8}a^{5}-\frac{113}{16}a^{4}-11a^{3}-\frac{49}{16}a^{2}+\frac{85}{4}a+\frac{45}{4}$, $\frac{7}{64}a^{9}-\frac{1}{4}a^{8}-\frac{41}{64}a^{7}+\frac{49}{32}a^{6}+\frac{95}{64}a^{5}-\frac{29}{16}a^{4}-\frac{265}{64}a^{3}-\frac{61}{32}a^{2}+\frac{135}{16}a+\frac{27}{8}$, $\frac{11}{32}a^{9}-\frac{3}{4}a^{8}-\frac{69}{32}a^{7}+\frac{73}{16}a^{6}+\frac{179}{32}a^{5}-\frac{51}{8}a^{4}-\frac{421}{32}a^{3}-\frac{45}{16}a^{2}+\frac{151}{8}a+\frac{51}{4}$, $\frac{31}{32}a^{9}-\frac{21}{8}a^{8}-\frac{153}{32}a^{7}+\frac{239}{16}a^{6}+\frac{279}{32}a^{5}-\frac{79}{4}a^{4}-\frac{937}{32}a^{3}+\frac{1}{16}a^{2}+\frac{435}{8}a+\frac{81}{4}$ | 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: | \( 11501.2800574 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 11501.2800574 \cdot 1}{2\cdot\sqrt{161960834367488}}\cr\approx \mathstrut & 1.79337358249 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 5 sibling: | 5.3.346112.1 |
Degree 6 sibling: | 6.0.58492928.4 |
Degree 10 sibling: | 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: | 5.3.346112.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
2.4.11.17 | $x^{4} + 8 x^{3} + 8 x + 2$ | $4$ | $1$ | $11$ | $D_{4}$ | $[3, 4]^{2}$ | |
2.4.11.14 | $x^{4} + 8 x + 10$ | $4$ | $1$ | $11$ | $D_{4}$ | $[3, 4]^{2}$ | |
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
13.3.2.3 | $x^{3} + 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
13.6.4.2 | $x^{6} - 156 x^{3} + 338$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |