Normalized defining polynomial
\( x^{9} - 7x^{7} - 7x^{6} - 7x^{5} + 84x^{4} + 35x^{3} - 280x^{2} + 28x + 280 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[5, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(45487052759281\) \(\medspace = 7^{8}\cdot 53^{4}\) | 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: | \(32.93\) | 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: | $7^{8/9}53^{1/2}\approx 41.05217698639053$ | ||
Ramified primes: | \(7\), \(53\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{6}a^{7}-\frac{1}{3}a^{6}-\frac{1}{6}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{6}a-\frac{1}{3}$, $\frac{1}{4631724}a^{8}+\frac{153313}{2315862}a^{7}+\frac{15377}{514636}a^{6}-\frac{963277}{4631724}a^{5}-\frac{244643}{1543908}a^{4}-\frac{7749}{257318}a^{3}+\frac{531719}{4631724}a^{2}+\frac{364169}{771954}a-\frac{351383}{1157931}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$
Unit group
Rank: | $6$ | 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: | $\frac{1426}{385977}a^{8}+\frac{2129}{771954}a^{7}-\frac{14488}{385977}a^{6}-\frac{3459}{257318}a^{5}-\frac{4079}{257318}a^{4}+\frac{46541}{257318}a^{3}+\frac{172466}{385977}a^{2}-\frac{1225327}{771954}a+\frac{115749}{128659}$, $\frac{25055}{4631724}a^{8}+\frac{14111}{2315862}a^{7}-\frac{60251}{1543908}a^{6}-\frac{535379}{4631724}a^{5}-\frac{215605}{1543908}a^{4}+\frac{123895}{257318}a^{3}+\frac{6013045}{4631724}a^{2}+\frac{272651}{771954}a-\frac{2081557}{1157931}$, $\frac{25055}{4631724}a^{8}+\frac{14111}{2315862}a^{7}-\frac{60251}{1543908}a^{6}-\frac{535379}{4631724}a^{5}-\frac{215605}{1543908}a^{4}+\frac{123895}{257318}a^{3}+\frac{6013045}{4631724}a^{2}-\frac{499303}{771954}a-\frac{2081557}{1157931}$, $\frac{608}{128659}a^{8}+\frac{1717}{128659}a^{7}-\frac{42}{128659}a^{6}-\frac{16648}{128659}a^{5}-\frac{39420}{128659}a^{4}-\frac{18775}{128659}a^{3}+\frac{93744}{128659}a^{2}+\frac{212996}{128659}a-\frac{139037}{128659}$, $\frac{10241}{4631724}a^{8}-\frac{76003}{2315862}a^{7}-\frac{2759}{514636}a^{6}+\frac{652363}{4631724}a^{5}+\frac{373721}{1543908}a^{4}+\frac{153753}{257318}a^{3}-\frac{10836593}{4631724}a^{2}-\frac{626999}{771954}a+\frac{4967969}{1157931}$, $\frac{112427}{4631724}a^{8}-\frac{27119}{1157931}a^{7}-\frac{131671}{1543908}a^{6}-\frac{144665}{4631724}a^{5}-\frac{529495}{1543908}a^{4}+\frac{233951}{128659}a^{3}-\frac{11353103}{4631724}a^{2}-\frac{228117}{128659}a+\frac{4441133}{1157931}$ | 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: | \( 1696.63643085 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{2}\cdot 1696.63643085 \cdot 3}{2\cdot\sqrt{45487052759281}}\cr\approx \mathstrut & 0.476700780441 \end{aligned}\]
Galois group
$C_3^3:S_4$ (as 9T30):
A solvable group of order 648 |
The 14 conjugacy class representatives for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
Character table for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
Intermediate fields
3.3.2597.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 27 siblings: | data not computed |
Degree 36 siblings: | data not computed |
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 | ${\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.9.0.1}{9} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.9.0.1}{9} }$ | R | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.9.8.1 | $x^{9} + 14$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ |
\(53\) | $\Q_{53}$ | $x + 51$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
53.2.1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
53.2.1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
53.4.2.1 | $x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.53.2t1.a.a | $1$ | $ 53 $ | \(\Q(\sqrt{53}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 2.2597.3t2.a.a | $2$ | $ 7^{2} \cdot 53 $ | 3.3.2597.1 | $S_3$ (as 3T2) | $1$ | $2$ |
3.137641.6t8.a.a | $3$ | $ 7^{2} \cdot 53^{2}$ | 4.0.2597.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.2597.4t5.a.a | $3$ | $ 7^{2} \cdot 53 $ | 4.0.2597.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
6.17515230173.18t217.a.a | $6$ | $ 7^{6} \cdot 53^{3}$ | 9.5.45487052759281.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $2$ | |
* | 6.17515230173.9t30.a.a | $6$ | $ 7^{6} \cdot 53^{3}$ | 9.5.45487052759281.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $2$ |
6.17515230173.36t1121.a.a | $6$ | $ 7^{6} \cdot 53^{3}$ | 9.5.45487052759281.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $-2$ | |
6.17515230173.36t1121.a.b | $6$ | $ 7^{6} \cdot 53^{3}$ | 9.5.45487052759281.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $-2$ | |
8.928307199169.12t177.a.a | $8$ | $ 7^{6} \cdot 53^{4}$ | 9.5.45487052759281.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
8.454...281.12t177.a.a | $8$ | $ 7^{8} \cdot 53^{4}$ | 9.5.45487052759281.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
8.454...281.12t178.a.a | $8$ | $ 7^{8} \cdot 53^{4}$ | 9.5.45487052759281.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
12.331...413.36t1123.a.a | $12$ | $ 7^{10} \cdot 53^{7}$ | 9.5.45487052759281.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
12.118...757.18t218.a.a | $12$ | $ 7^{10} \cdot 53^{5}$ | 9.5.45487052759281.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ |