Normalized defining polynomial
\( x^{9} - 819x^{7} + 223587x^{5} - 22607130x^{3} + 617174649x - 1016567279 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[9, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(17820338848416865911969\) \(\medspace = 3^{22}\cdot 7^{6}\cdot 13^{6}\) | 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: | \(296.70\) | 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))
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Galois root discriminant: | $3^{22/9}7^{2/3}13^{2/3}\approx 296.70440583414893$ | ||
Ramified primes: | \(3\), \(7\), \(13\) | 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: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $9$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(2457=3^{3}\cdot 7\cdot 13\) | ||
Dirichlet character group: | $\lbrace$$\chi_{2457}(256,·)$, $\chi_{2457}(1,·)$, $\chi_{2457}(835,·)$, $\chi_{2457}(1894,·)$, $\chi_{2457}(1639,·)$, $\chi_{2457}(16,·)$, $\chi_{2457}(1075,·)$, $\chi_{2457}(820,·)$, $\chi_{2457}(1654,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{91}a^{3}$, $\frac{1}{91}a^{4}$, $\frac{1}{57421}a^{5}+\frac{107}{57421}a^{4}+\frac{176}{57421}a^{3}+\frac{203}{631}a^{2}-\frac{176}{631}a-\frac{87}{631}$, $\frac{1}{5225311}a^{6}-\frac{6}{57421}a^{4}+\frac{107}{57421}a^{3}+\frac{9}{631}a^{2}+\frac{310}{631}a-\frac{182}{631}$, $\frac{1}{5225311}a^{7}+\frac{118}{57421}a^{4}-\frac{18}{57421}a^{3}+\frac{266}{631}a^{2}+\frac{24}{631}a+\frac{109}{631}$, $\frac{1}{5225311}a^{8}-\frac{24}{57421}a^{4}+\frac{283}{57421}a^{3}+\frac{48}{631}a^{2}+\frac{54}{631}a+\frac{170}{631}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}\times C_{3}$, which has order $9$ (assuming GRH)
Unit group
Rank: | $8$ | 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)
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Fundamental units: | $\frac{5}{5225311}a^{6}-\frac{30}{57421}a^{4}-\frac{96}{57421}a^{3}+\frac{45}{631}a^{2}+\frac{288}{631}a-\frac{279}{631}$, $\frac{6}{5225311}a^{6}-\frac{36}{57421}a^{4}+\frac{11}{57421}a^{3}+\frac{54}{631}a^{2}-\frac{33}{631}a-\frac{461}{631}$, $\frac{522}{5225311}a^{8}-\frac{9622}{5225311}a^{7}-\frac{250399}{5225311}a^{6}+\frac{50801}{57421}a^{5}+\frac{346540}{57421}a^{4}-\frac{6416992}{57421}a^{3}-\frac{123465}{631}a^{2}+\frac{2291934}{631}a-\frac{3466648}{631}$, $\frac{3313}{5225311}a^{8}+\frac{45358}{5225311}a^{7}-\frac{2095953}{5225311}a^{6}-\frac{315450}{57421}a^{5}+\frac{295961}{4417}a^{4}+\frac{52735037}{57421}a^{3}-\frac{1163453}{631}a^{2}-\frac{15986849}{631}a+\frac{29924443}{631}$, $\frac{124}{401947}a^{8}-\frac{2480}{746473}a^{7}-\frac{1133147}{5225311}a^{6}+\frac{134072}{57421}a^{5}+\frac{193568}{4417}a^{4}-\frac{27088364}{57421}a^{3}-\frac{1195230}{631}a^{2}+\frac{12865690}{631}a-\frac{18379182}{631}$, $\frac{142}{5225311}a^{8}+\frac{57}{106639}a^{7}-\frac{62987}{5225311}a^{6}-\frac{13989}{57421}a^{5}+\frac{11600}{8203}a^{4}+\frac{1767008}{57421}a^{3}-\frac{13637}{631}a^{2}-\frac{500545}{631}a+\frac{605056}{631}$, $\frac{2}{30919}a^{8}-\frac{5730}{5225311}a^{7}-\frac{174364}{5225311}a^{6}+\frac{4630}{8203}a^{5}+\frac{247615}{57421}a^{4}-\frac{4203944}{57421}a^{3}-\frac{85868}{631}a^{2}+\frac{1551582}{631}a-\frac{2350682}{631}$, $\frac{1959}{5225311}a^{8}-\frac{34057}{5225311}a^{7}-\frac{1012356}{5225311}a^{6}+\frac{27629}{8203}a^{5}+\frac{1451066}{57421}a^{4}-\frac{25226627}{57421}a^{3}-\frac{528866}{631}a^{2}+\frac{9194237}{631}a-\frac{13833459}{631}$ (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)]
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Regulator: | \( 24786188.8419 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 24786188.8419 \cdot 9}{2\cdot\sqrt{17820338848416865911969}}\cr\approx \mathstrut & 0.427793567528 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 9 |
The 9 conjugacy class representatives for $C_9$ |
Character table for $C_9$ |
Intermediate fields
\(\Q(\zeta_{9})^+\) |
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 | ${\href{/padicField/2.9.0.1}{9} }$ | R | ${\href{/padicField/5.9.0.1}{9} }$ | R | ${\href{/padicField/11.9.0.1}{9} }$ | R | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.1.0.1}{1} }^{9}$ | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.3.0.1}{3} }^{3}$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.9.22.6 | $x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
\(7\) | 7.9.6.2 | $x^{9} + 252 x^{6} - 2352 x^{3} + 5488$ | $3$ | $3$ | $6$ | $C_9$ | $[\ ]_{3}^{3}$ |
\(13\) | 13.9.6.3 | $x^{9} + 338 x^{3} - 24167$ | $3$ | $3$ | $6$ | $C_9$ | $[\ ]_{3}^{3}$ |