Normalized defining polynomial
\( x^{5} - x^{4} - 228x^{3} - 868x^{2} + 3056x + 7552 \)
Invariants
Degree: | $5$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[5, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(106302733681\) \(\medspace = 571^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(160.44\) | 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: | $571^{4/5}\approx 160.43860848150953$ | ||
Ramified primes: | \(571\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $5$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(571\) | ||
Dirichlet character group: | $\lbrace$$\chi_{571}(481,·)$, $\chi_{571}(106,·)$, $\chi_{571}(387,·)$, $\chi_{571}(167,·)$, $\chi_{571}(1,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{3}+\frac{1}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{3968}a^{4}-\frac{235}{3968}a^{3}-\frac{395}{1984}a^{2}+\frac{183}{496}a+\frac{27}{62}$
Monogenic: | No | |
Index: | $32$ | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | 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: | $\frac{1217543}{1984}a^{4}-\frac{3573725}{1984}a^{3}-\frac{135347685}{992}a^{2}-\frac{66638491}{248}a+\frac{74256770}{31}$, $\frac{30493}{1984}a^{4}+\frac{458161}{1984}a^{3}+\frac{477033}{992}a^{2}-\frac{729273}{248}a-\frac{187010}{31}$, $\frac{2996025}{1984}a^{4}-\frac{8792915}{1984}a^{3}-\frac{333040851}{992}a^{2}-\frac{163973753}{248}a+\frac{182718096}{31}$, $\frac{491055}{1984}a^{4}+\frac{1136779}{1984}a^{3}-\frac{54081581}{992}a^{2}-\frac{98095267}{248}a-\frac{17302028}{31}$ | 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: | \( 148439.26957 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{0}\cdot 148439.26957 \cdot 1}{2\cdot\sqrt{106302733681}}\cr\approx \mathstrut & 7.2844467816 \end{aligned}\]
Galois group
A cyclic group of order 5 |
The 5 conjugacy class representatives for $C_5$ |
Character table for $C_5$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.1.0.1}{1} }^{5}$ | ${\href{/padicField/3.5.0.1}{5} }$ | ${\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.5.0.1}{5} }$ | ${\href{/padicField/13.5.0.1}{5} }$ | ${\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.5.0.1}{5} }$ | ${\href{/padicField/29.1.0.1}{1} }^{5}$ | ${\href{/padicField/31.1.0.1}{1} }^{5}$ | ${\href{/padicField/37.5.0.1}{5} }$ | ${\href{/padicField/41.1.0.1}{1} }^{5}$ | ${\href{/padicField/43.5.0.1}{5} }$ | ${\href{/padicField/47.1.0.1}{1} }^{5}$ | ${\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.1.0.1}{1} }^{5}$ |
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 |
---|---|---|---|---|---|---|---|
\(571\) | Deg $5$ | $5$ | $1$ | $4$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.571.5t1.a.a | $1$ | $ 571 $ | 5.5.106302733681.1 | $C_5$ (as 5T1) | $0$ | $1$ |
* | 1.571.5t1.a.b | $1$ | $ 571 $ | 5.5.106302733681.1 | $C_5$ (as 5T1) | $0$ | $1$ |
* | 1.571.5t1.a.c | $1$ | $ 571 $ | 5.5.106302733681.1 | $C_5$ (as 5T1) | $0$ | $1$ |
* | 1.571.5t1.a.d | $1$ | $ 571 $ | 5.5.106302733681.1 | $C_5$ (as 5T1) | $0$ | $1$ |