Normalized defining polynomial
\( x^{10} - 4x^{9} - 12x^{8} + 52x^{7} + 32x^{6} - 165x^{5} - 38x^{4} + 184x^{3} + 36x^{2} - 64x - 17 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[10, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(6438696919952653\) \(\medspace = 186037^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(38.10\) | 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))
| |
Galois root discriminant: | $186037^{1/2}\approx 431.3200667717652$ | ||
Ramified primes: | \(186037\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{186037}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{38123}a^{9}-\frac{11815}{38123}a^{8}+\frac{16773}{38123}a^{7}-\frac{18743}{38123}a^{6}-\frac{6656}{38123}a^{5}+\frac{4225}{38123}a^{4}+\frac{1494}{38123}a^{3}+\frac{5499}{38123}a^{2}+\frac{12939}{38123}a+\frac{12514}{38123}$
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)
| |
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: | $\frac{8034}{38123}a^{9}-\frac{33563}{38123}a^{8}-\frac{86769}{38123}a^{7}+\frac{423941}{38123}a^{6}+\frac{126634}{38123}a^{5}-\frac{1243879}{38123}a^{4}+\frac{146543}{38123}a^{3}+\frac{1252468}{38123}a^{2}-\frac{200110}{38123}a-\frac{412228}{38123}$, $\frac{8034}{38123}a^{9}-\frac{33563}{38123}a^{8}-\frac{86769}{38123}a^{7}+\frac{423941}{38123}a^{6}+\frac{126634}{38123}a^{5}-\frac{1243879}{38123}a^{4}+\frac{146543}{38123}a^{3}+\frac{1252468}{38123}a^{2}-\frac{200110}{38123}a-\frac{374105}{38123}$, $\frac{12000}{38123}a^{9}-\frac{38686}{38123}a^{8}-\frac{165932}{38123}a^{7}+\frac{467176}{38123}a^{6}+\frac{643776}{38123}a^{5}-\frac{1147280}{38123}a^{4}-\frac{1019131}{38123}a^{3}+\frac{645178}{38123}a^{2}+\frac{450497}{38123}a+\frac{1503}{38123}$, $\frac{3408}{38123}a^{9}-\frac{7632}{38123}a^{8}-\frac{60239}{38123}a^{7}+\frac{94250}{38123}a^{6}+\frac{342644}{38123}a^{5}-\frac{240432}{38123}a^{4}-\frac{703144}{38123}a^{3}+\frac{98445}{38123}a^{2}+\frac{445277}{38123}a+\frac{102444}{38123}$, $\frac{3408}{38123}a^{9}-\frac{7632}{38123}a^{8}-\frac{60239}{38123}a^{7}+\frac{94250}{38123}a^{6}+\frac{342644}{38123}a^{5}-\frac{240432}{38123}a^{4}-\frac{703144}{38123}a^{3}+\frac{98445}{38123}a^{2}+\frac{483400}{38123}a+\frac{102444}{38123}$, $\frac{7870}{38123}a^{9}-\frac{40176}{38123}a^{8}-\frac{54562}{38123}a^{7}+\frac{486076}{38123}a^{6}-\frac{230456}{38123}a^{5}-\frac{1250565}{38123}a^{4}+\frac{930848}{38123}a^{3}+\frac{998723}{38123}a^{2}-\frac{606571}{38123}a-\frac{291513}{38123}$, $\frac{12000}{38123}a^{9}-\frac{38686}{38123}a^{8}-\frac{165932}{38123}a^{7}+\frac{467176}{38123}a^{6}+\frac{643776}{38123}a^{5}-\frac{1147280}{38123}a^{4}-\frac{1019131}{38123}a^{3}+\frac{645178}{38123}a^{2}+\frac{450497}{38123}a-\frac{36620}{38123}$, $\frac{8128}{38123}a^{9}-\frac{38606}{38123}a^{8}-\frac{73150}{38123}a^{7}+\frac{492003}{38123}a^{6}-\frac{41554}{38123}a^{5}-\frac{1494820}{38123}a^{4}+\frac{515717}{38123}a^{3}+\frac{1655005}{38123}a^{2}-\frac{432518}{38123}a-\frac{646463}{38123}$, $\frac{14190}{38123}a^{9}-\frac{28019}{38123}a^{8}-\frac{259880}{38123}a^{7}+\frac{325985}{38123}a^{6}+\frac{1545074}{38123}a^{5}-\frac{624697}{38123}a^{4}-\frac{3275106}{38123}a^{3}-\frac{273832}{38123}a^{2}+\frac{1986438}{38123}a+\frac{568571}{38123}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 56496.3603639 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{0}\cdot 56496.3603639 \cdot 1}{2\cdot\sqrt{6438696919952653}}\cr\approx \mathstrut & 0.360488520981 \end{aligned}\]
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: | 5.5.186037.1 |
Degree 6 sibling: | 6.6.6438696919952653.1 |
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: | 5.5.186037.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(186037\) | $\Q_{186037}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $2$ | $3$ | $3$ |