Normalized defining polynomial
\( x^{10} - 4x^{8} - x^{7} + 13x^{6} - 2x^{5} - 30x^{4} + 22x^{3} - 2x^{2} - 3x + 1 \)
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: | \(-399826899863\) \(\medspace = -\,53^{3}\cdot 139^{3}\) | 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: | \(14.46\) | 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: | $53^{1/2}139^{1/2}\approx 85.83122974768565$ | ||
Ramified primes: | \(53\), \(139\) | 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: | \(\Q(\sqrt{-7367}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5023}a^{9}-\frac{313}{5023}a^{8}-\frac{2495}{5023}a^{7}+\frac{2369}{5023}a^{6}+\frac{1920}{5023}a^{5}+\frac{1798}{5023}a^{4}-\frac{228}{5023}a^{3}+\frac{1064}{5023}a^{2}-\frac{1516}{5023}a+\frac{2343}{5023}$
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)
| |
Fundamental units: | $\frac{5787}{5023}a^{9}+\frac{1972}{5023}a^{8}-\frac{22555}{5023}a^{7}-\frac{13433}{5023}a^{6}+\frac{70486}{5023}a^{5}+\frac{12439}{5023}a^{4}-\frac{169169}{5023}a^{3}+\frac{69492}{5023}a^{2}+\frac{12135}{5023}a-\frac{13205}{5023}$, $a$, $\frac{5464}{5023}a^{9}+\frac{2611}{5023}a^{8}-\frac{20350}{5023}a^{7}-\frac{15124}{5023}a^{6}+\frac{63132}{5023}a^{5}+\frac{19376}{5023}a^{4}-\frac{150778}{5023}a^{3}+\frac{47292}{5023}a^{2}+\frac{4526}{5023}a-\frac{6498}{5023}$, $\frac{764}{5023}a^{9}+\frac{1972}{5023}a^{8}-\frac{2463}{5023}a^{7}-\frac{8410}{5023}a^{6}+\frac{5187}{5023}a^{5}+\frac{22485}{5023}a^{4}-\frac{18479}{5023}a^{3}-\frac{41014}{5023}a^{2}+\frac{22181}{5023}a+\frac{1864}{5023}$, $\frac{6345}{5023}a^{9}+\frac{3123}{5023}a^{8}-\frac{23394}{5023}a^{7}-\frac{17603}{5023}a^{6}+\frac{71947}{5023}a^{5}+\frac{21169}{5023}a^{4}-\frac{175841}{5023}a^{3}+\frac{55421}{5023}a^{2}+\frac{5048}{5023}a-\frac{11791}{5023}$, $\frac{4735}{5023}a^{9}-\frac{270}{5023}a^{8}-\frac{19821}{5023}a^{7}-\frac{4167}{5023}a^{6}+\frac{64869}{5023}a^{5}-\frac{10501}{5023}a^{4}-\frac{150325}{5023}a^{3}+\frac{110477}{5023}a^{2}+\frac{9653}{5023}a-\frac{21794}{5023}$ | 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: | \( 90.8963525708 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 90.8963525708 \cdot 1}{2\cdot\sqrt{399826899863}}\cr\approx \mathstrut & 0.285259452134 \end{aligned}\]
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: | 5.3.7367.1 |
Degree 6 sibling: | 6.0.399826899863.1 |
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: | 5.3.7367.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.3.0.1}{3} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.3.0.1}{3} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | R | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(53\) | $\Q_{53}$ | $x + 51$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{53}$ | $x + 51$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
53.2.1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
53.4.2.1 | $x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(139\) | $\Q_{139}$ | $x + 137$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{139}$ | $x + 137$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
139.2.0.1 | $x^{2} + 138 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
139.2.1.2 | $x^{2} + 139$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
139.4.2.1 | $x^{4} + 276 x^{3} + 19326 x^{2} + 38916 x + 2665885$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |