Normalized defining polynomial
\( x^{10} - x^{9} + 126 x^{8} + 140 x^{7} + 3959 x^{6} + 9931 x^{5} + 51875 x^{4} + 157846 x^{3} + \cdots + 795857 \)
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: | \(-37540482602183844047\) \(\medspace = -\,17^{5}\cdot 31^{9}\) | 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: | \(90.67\) | 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: | $17^{1/2}31^{9/10}\approx 90.66715431291931$ | ||
Ramified primes: | \(17\), \(31\) | 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: | \(\Q(\sqrt{-527}) \) | ||
$\card{ \Gal(K/\Q) }$: | $10$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(527=17\cdot 31\) | ||
Dirichlet character group: | $\lbrace$$\chi_{527}(256,·)$, $\chi_{527}(1,·)$, $\chi_{527}(35,·)$, $\chi_{527}(356,·)$, $\chi_{527}(171,·)$, $\chi_{527}(492,·)$, $\chi_{527}(526,·)$, $\chi_{527}(271,·)$, $\chi_{527}(339,·)$, $\chi_{527}(188,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-527}) \), 10.0.37540482602183844047.2$^{15}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{22\!\cdots\!11}a^{9}+\frac{52\!\cdots\!61}{22\!\cdots\!11}a^{8}-\frac{43\!\cdots\!80}{22\!\cdots\!11}a^{7}-\frac{14\!\cdots\!26}{22\!\cdots\!11}a^{6}+\frac{99\!\cdots\!25}{22\!\cdots\!11}a^{5}-\frac{71\!\cdots\!82}{22\!\cdots\!11}a^{4}+\frac{78\!\cdots\!62}{22\!\cdots\!11}a^{3}+\frac{14\!\cdots\!46}{22\!\cdots\!11}a^{2}+\frac{10\!\cdots\!05}{22\!\cdots\!11}a+\frac{10\!\cdots\!84}{22\!\cdots\!11}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{738}$, which has order $738$ (assuming GRH)
Unit group
Rank: | $4$ | 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: | $\frac{17\!\cdots\!65}{22\!\cdots\!11}a^{9}-\frac{27\!\cdots\!30}{22\!\cdots\!11}a^{8}+\frac{20\!\cdots\!95}{22\!\cdots\!11}a^{7}+\frac{13\!\cdots\!70}{22\!\cdots\!11}a^{6}+\frac{55\!\cdots\!69}{22\!\cdots\!11}a^{5}+\frac{13\!\cdots\!45}{22\!\cdots\!11}a^{4}+\frac{56\!\cdots\!75}{22\!\cdots\!11}a^{3}+\frac{19\!\cdots\!05}{22\!\cdots\!11}a^{2}+\frac{16\!\cdots\!69}{22\!\cdots\!11}a+\frac{11\!\cdots\!62}{22\!\cdots\!11}$, $\frac{19\!\cdots\!33}{22\!\cdots\!11}a^{9}+\frac{13\!\cdots\!96}{22\!\cdots\!11}a^{8}+\frac{21\!\cdots\!17}{22\!\cdots\!11}a^{7}+\frac{68\!\cdots\!55}{22\!\cdots\!11}a^{6}+\frac{50\!\cdots\!75}{22\!\cdots\!11}a^{5}+\frac{24\!\cdots\!59}{22\!\cdots\!11}a^{4}+\frac{58\!\cdots\!57}{22\!\cdots\!11}a^{3}+\frac{14\!\cdots\!47}{22\!\cdots\!11}a^{2}+\frac{28\!\cdots\!12}{22\!\cdots\!11}a+\frac{11\!\cdots\!20}{22\!\cdots\!11}$, $\frac{15\!\cdots\!18}{22\!\cdots\!11}a^{9}-\frac{18\!\cdots\!23}{22\!\cdots\!11}a^{8}+\frac{17\!\cdots\!30}{22\!\cdots\!11}a^{7}+\frac{22\!\cdots\!30}{22\!\cdots\!11}a^{6}+\frac{36\!\cdots\!05}{22\!\cdots\!11}a^{5}+\frac{12\!\cdots\!80}{22\!\cdots\!11}a^{4}+\frac{33\!\cdots\!32}{22\!\cdots\!11}a^{3}+\frac{10\!\cdots\!51}{22\!\cdots\!11}a^{2}+\frac{12\!\cdots\!73}{22\!\cdots\!11}a-\frac{30\!\cdots\!18}{22\!\cdots\!11}$, $\frac{95\!\cdots\!57}{22\!\cdots\!11}a^{9}-\frac{68\!\cdots\!17}{22\!\cdots\!11}a^{8}+\frac{12\!\cdots\!30}{22\!\cdots\!11}a^{7}+\frac{16\!\cdots\!34}{22\!\cdots\!11}a^{6}+\frac{37\!\cdots\!78}{22\!\cdots\!11}a^{5}+\frac{10\!\cdots\!60}{22\!\cdots\!11}a^{4}+\frac{43\!\cdots\!73}{22\!\cdots\!11}a^{3}+\frac{14\!\cdots\!99}{22\!\cdots\!11}a^{2}+\frac{14\!\cdots\!80}{22\!\cdots\!11}a+\frac{51\!\cdots\!90}{22\!\cdots\!11}$ (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: | \( 485.913224212 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 485.913224212 \cdot 738}{2\cdot\sqrt{37540482602183844047}}\cr\approx \mathstrut & 0.286572492393 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 10 |
The 10 conjugacy class representatives for $C_{10}$ |
Character table for $C_{10}$ |
Intermediate fields
\(\Q(\sqrt{-527}) \), 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 | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{5}$ | ${\href{/padicField/7.10.0.1}{10} }$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }$ | R | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/37.1.0.1}{1} }^{10}$ | ${\href{/padicField/41.10.0.1}{10} }$ | ${\href{/padicField/43.10.0.1}{10} }$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(17\) | 17.10.5.2 | $x^{10} + 83521 x^{2} - 19877998$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
\(31\) | 31.10.9.1 | $x^{10} + 186$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |