Normalized defining polynomial
\( x^{5} - x^{4} - 164x^{3} - 27559x^{2} - 210375x - 5467886 \)
Invariants
Degree: | $5$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1165308928144589\) \(\medspace = 137^{4}\cdot 149^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(1031.07\) | 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: | $137^{4/5}149^{5/6}\approx 3314.095436721782$ | ||
Ramified primes: | \(137\), \(149\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{149}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{52023904}a^{4}+\frac{527127}{3251494}a^{3}-\frac{1003789}{13005976}a^{2}+\frac{13622917}{52023904}a-\frac{6640809}{26011952}$
Monogenic: | No | |
Index: | $2$ | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $2$ | 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: | $\frac{30\!\cdots\!23}{26011952}a^{4}-\frac{21\!\cdots\!40}{1625747}a^{3}-\frac{24\!\cdots\!07}{6502988}a^{2}-\frac{68\!\cdots\!93}{26011952}a+\frac{18\!\cdots\!41}{13005976}$, $\frac{12\!\cdots\!07}{26011952}a^{4}-\frac{67\!\cdots\!37}{1625747}a^{3}-\frac{10\!\cdots\!27}{6502988}a^{2}-\frac{28\!\cdots\!61}{26011952}a+\frac{36\!\cdots\!93}{13005976}$ (assuming GRH) | 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: | \( 3503103.6972046597 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{2}\cdot 3503103.6972046597 \cdot 1}{2\cdot\sqrt{1165308928144589}}\cr\approx \mathstrut & 4.05127937536126 \end{aligned}\] (assuming GRH)
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 6 sibling: | deg 6 |
Degree 10 siblings: | deg 10, deg 10 |
Degree 12 sibling: | deg 12 |
Degree 15 sibling: | deg 15 |
Degree 20 siblings: | deg 20, deg 20, deg 20 |
Degree 24 sibling: | deg 24 |
Degree 30 siblings: | deg 30, deg 30, deg 30 |
Degree 40 sibling: | deg 40 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{3}$ | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(137\) | 137.5.4.1 | $x^{5} + 137$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ |
\(149\) | 149.2.1.2 | $x^{2} + 298$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
149.3.2.1 | $x^{3} + 149$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |