Normalized defining polynomial
\( x^{10} - x^{9} + 2x^{8} + 16x^{7} - 9x^{6} + 11x^{5} + 43x^{4} - 6x^{3} + 63x^{2} - 20x + 25 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-26439622160671\) \(\medspace = -\,31^{9}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(21.99\) | 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: | $31^{9/10}\approx 21.990014941549536$ | ||
Ramified primes: | \(31\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-31}) \) | ||
$\card{ \Gal(K/\Q) }$: | $10$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(31\) | ||
Dirichlet character group: | $\lbrace$$\chi_{31}(1,·)$, $\chi_{31}(2,·)$, $\chi_{31}(4,·)$, $\chi_{31}(8,·)$, $\chi_{31}(15,·)$, $\chi_{31}(16,·)$, $\chi_{31}(23,·)$, $\chi_{31}(27,·)$, $\chi_{31}(29,·)$, $\chi_{31}(30,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-31}) \), 10.0.26439622160671.1$^{15}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5}a^{5}-\frac{1}{5}a$, $\frac{1}{5}a^{6}-\frac{1}{5}a^{2}$, $\frac{1}{5}a^{7}-\frac{1}{5}a^{3}$, $\frac{1}{5}a^{8}-\frac{1}{5}a^{4}$, $\frac{1}{8375}a^{9}+\frac{628}{8375}a^{8}-\frac{286}{8375}a^{7}-\frac{653}{8375}a^{6}-\frac{371}{8375}a^{5}-\frac{3873}{8375}a^{4}+\frac{2726}{8375}a^{3}+\frac{2798}{8375}a^{2}+\frac{251}{1675}a-\frac{49}{335}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $5$ |
Class group and class number
$C_{3}$, which has order $3$
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: | $\frac{93}{8375}a^{9}-\frac{221}{8375}a^{8}+\frac{202}{8375}a^{7}+\frac{1246}{8375}a^{6}-\frac{2678}{8375}a^{5}-\frac{64}{8375}a^{4}+\frac{593}{8375}a^{3}-\frac{2761}{8375}a^{2}+\frac{228}{1675}a-\frac{202}{335}$, $\frac{133}{8375}a^{9}-\frac{226}{8375}a^{8}+\frac{487}{8375}a^{7}+\frac{1926}{8375}a^{6}-\frac{2443}{8375}a^{5}+\frac{4141}{8375}a^{4}+\frac{5783}{8375}a^{3}-\frac{1391}{8375}a^{2}+\frac{553}{1675}a+\frac{518}{335}$, $\frac{617}{8375}a^{9}-\frac{1124}{8375}a^{8}+\frac{1088}{8375}a^{7}+\frac{9149}{8375}a^{6}-\frac{14507}{8375}a^{5}+\frac{584}{8375}a^{4}+\frac{22017}{8375}a^{3}-\frac{25684}{8375}a^{2}+\frac{3112}{1675}a-\frac{2093}{335}$, $\frac{681}{8375}a^{9}-\frac{1132}{8375}a^{8}+\frac{1209}{8375}a^{7}+\frac{10907}{8375}a^{6}-\frac{14801}{8375}a^{5}+\frac{2287}{8375}a^{4}+\frac{35681}{8375}a^{3}-\frac{32537}{8375}a^{2}+\frac{4436}{1675}a-\frac{204}{335}$ | 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: | \( 485.913224212 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 485.913224212 \cdot 3}{2\cdot\sqrt{26439622160671}}\cr\approx \mathstrut & 1.38810301402 \end{aligned}\]
Galois group
A cyclic group of order 10 |
The 10 conjugacy class representatives for $C_{10}$ |
Character table for $C_{10}$ |
Intermediate fields
\(\Q(\sqrt{-31}) \), 5.5.923521.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }$ | ${\href{/padicField/5.1.0.1}{1} }^{10}$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }$ | ${\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }$ | ${\href{/padicField/29.10.0.1}{10} }$ | R | ${\href{/padicField/37.2.0.1}{2} }^{5}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(31\) | 31.10.9.8 | $x^{10} + 31$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.31.2t1.a.a | $1$ | $ 31 $ | \(\Q(\sqrt{-31}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.31.5t1.a.a | $1$ | $ 31 $ | 5.5.923521.1 | $C_5$ (as 5T1) | $0$ | $1$ |
* | 1.31.10t1.a.a | $1$ | $ 31 $ | 10.0.26439622160671.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ |
* | 1.31.5t1.a.b | $1$ | $ 31 $ | 5.5.923521.1 | $C_5$ (as 5T1) | $0$ | $1$ |
* | 1.31.10t1.a.b | $1$ | $ 31 $ | 10.0.26439622160671.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ |
* | 1.31.5t1.a.c | $1$ | $ 31 $ | 5.5.923521.1 | $C_5$ (as 5T1) | $0$ | $1$ |
* | 1.31.10t1.a.c | $1$ | $ 31 $ | 10.0.26439622160671.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ |
* | 1.31.5t1.a.d | $1$ | $ 31 $ | 5.5.923521.1 | $C_5$ (as 5T1) | $0$ | $1$ |
* | 1.31.10t1.a.d | $1$ | $ 31 $ | 10.0.26439622160671.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ |