Normalized defining polynomial
\( x^{10} - 7x^{8} - 10x^{7} + 11x^{6} + 10x^{5} - 17x^{4} - 19x^{3} - 2x^{2} + 4x + 1 \)
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: | \(-11135879290691\) \(\medspace = -\,137^{3}\cdot 163^{3}\) | 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: | \(20.17\) | 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: | $137^{1/2}163^{1/2}\approx 149.4356048604214$ | ||
Ramified primes: | \(137\), \(163\) | 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: | \(\Q(\sqrt{-22331}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{1177}a^{9}-\frac{333}{1177}a^{8}+\frac{244}{1177}a^{7}-\frac{49}{1177}a^{6}-\frac{150}{1177}a^{5}+\frac{526}{1177}a^{4}+\frac{18}{107}a^{3}-\frac{41}{1177}a^{2}-\frac{43}{107}a-\frac{205}{1177}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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: | $a^{9}-7a^{7}-10a^{6}+11a^{5}+10a^{4}-17a^{3}-19a^{2}-2a+4$, $\frac{1167}{1177}a^{9}-\frac{201}{1177}a^{8}-\frac{8325}{1177}a^{7}-\frac{10103}{1177}a^{6}+\frac{15624}{1177}a^{5}+\frac{10041}{1177}a^{4}-\frac{2106}{107}a^{3}-\frac{16068}{1177}a^{2}+\frac{216}{107}a+\frac{4404}{1177}$, $\frac{129}{1177}a^{9}+\frac{592}{1177}a^{8}-\frac{1480}{1177}a^{7}-\frac{5144}{1177}a^{6}-\frac{518}{1177}a^{5}+\frac{11358}{1177}a^{4}-\frac{460}{107}a^{3}-\frac{13528}{1177}a^{2}+\frac{124}{107}a+\frac{2980}{1177}$, $\frac{2868}{1177}a^{9}-\frac{1674}{1177}a^{8}-\frac{19355}{1177}a^{7}-\frac{16947}{1177}a^{6}+\frac{42954}{1177}a^{5}+\frac{3185}{1177}a^{4}-\frac{5086}{107}a^{3}-\frac{17543}{1177}a^{2}+\frac{903}{107}a+\frac{2914}{1177}$, $\frac{3367}{1177}a^{9}-\frac{1884}{1177}a^{8}-\frac{22361}{1177}a^{7}-\frac{21389}{1177}a^{6}+\frac{48140}{1177}a^{5}+\frac{6719}{1177}a^{4}-\frac{5306}{107}a^{3}-\frac{33294}{1177}a^{2}+\frac{953}{107}a+\frac{7726}{1177}$, $\frac{1403}{1177}a^{9}+\frac{70}{1177}a^{8}-\frac{9591}{1177}a^{7}-\frac{14605}{1177}a^{6}+\frac{13180}{1177}a^{5}+\frac{12946}{1177}a^{4}-\frac{1817}{107}a^{3}-\frac{25744}{1177}a^{2}-\frac{623}{107}a+\frac{750}{1177}$ | 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: | \( 788.288964051 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 788.288964051 \cdot 1}{2\cdot\sqrt{11135879290691}}\cr\approx \mathstrut & 0.468762413105 \end{aligned}\]
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: | 5.3.22331.1 |
Degree 6 sibling: | 6.0.11135879290691.1 |
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: | 5.3.22331.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.3.0.1}{3} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(137\) | $\Q_{137}$ | $x + 134$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{137}$ | $x + 134$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
137.2.1.1 | $x^{2} + 137$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
137.4.2.1 | $x^{4} + 20538 x^{3} + 106780719 x^{2} + 13640908302 x + 496637065$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(163\) | $\Q_{163}$ | $x + 161$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{163}$ | $x + 161$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
163.2.0.1 | $x^{2} + 159 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
163.2.1.2 | $x^{2} + 163$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
163.4.2.1 | $x^{4} + 318 x^{3} + 25611 x^{2} + 52470 x + 4146724$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |