Normalized defining polynomial
\( x^{7} - x^{6} - 468x^{5} - 2543x^{4} + 23004x^{3} + 169598x^{2} + 130839x - 450131 \)
Invariants
Degree: | $7$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[7, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1704986606307341449\) \(\medspace = 1093^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(402.28\) | 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: | $1093^{6/7}\approx 402.2828955083758$ | ||
Ramified primes: | \(1093\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $7$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(1093\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1093}(1,·)$, $\chi_{1093}(3,·)$, $\chi_{1093}(243,·)$, $\chi_{1093}(81,·)$, $\chi_{1093}(9,·)$, $\chi_{1093}(729,·)$, $\chi_{1093}(27,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{13}a^{5}+\frac{4}{13}a^{4}+\frac{1}{13}a^{3}-\frac{1}{13}a^{2}-\frac{4}{13}a-\frac{1}{13}$, $\frac{1}{542269032877}a^{6}+\frac{4493475598}{542269032877}a^{5}+\frac{45154750097}{542269032877}a^{4}+\frac{96153868131}{542269032877}a^{3}+\frac{148625348572}{542269032877}a^{2}-\frac{165291821411}{542269032877}a+\frac{18564464703}{49297184807}$
Monogenic: | No | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{96018492}{49297184807}a^{6}+\frac{372694710}{49297184807}a^{5}-\frac{40816531994}{49297184807}a^{4}-\frac{435165338925}{49297184807}a^{3}-\frac{883728551756}{49297184807}a^{2}+\frac{1634880866896}{49297184807}a+\frac{214309707476}{49297184807}$, $\frac{1412092195}{542269032877}a^{6}+\frac{34105200361}{542269032877}a^{5}+\frac{152969221695}{542269032877}a^{4}-\frac{1085014384647}{542269032877}a^{3}-\frac{7937568788173}{542269032877}a^{2}-\frac{5881544354482}{542269032877}a+\frac{1878597093579}{49297184807}$, $\frac{6658040815}{542269032877}a^{6}+\frac{1431280533}{542269032877}a^{5}-\frac{3113178983504}{542269032877}a^{4}-\frac{20712355139207}{542269032877}a^{3}+\frac{127563935489087}{542269032877}a^{2}+\frac{12\!\cdots\!15}{542269032877}a+\frac{221466306344754}{49297184807}$, $\frac{37568079873}{542269032877}a^{6}-\frac{608904535193}{542269032877}a^{5}-\frac{8317241812135}{542269032877}a^{4}+\frac{30983890108742}{542269032877}a^{3}+\frac{392771260386017}{542269032877}a^{2}+\frac{396191407833321}{542269032877}a-\frac{101053151523442}{49297184807}$, $\frac{103037617}{542269032877}a^{6}+\frac{489719501}{542269032877}a^{5}-\frac{47080417510}{542269032877}a^{4}-\frac{318636070036}{542269032877}a^{3}+\frac{3356078297327}{542269032877}a^{2}+\frac{14818485877919}{542269032877}a-\frac{2487979713442}{49297184807}$, $\frac{978163405}{41713002529}a^{6}-\frac{109469120500}{542269032877}a^{5}-\frac{5114998279348}{542269032877}a^{4}+\frac{6561193186445}{542269032877}a^{3}+\frac{241408806396588}{542269032877}a^{2}+\frac{317814619794720}{542269032877}a-\frac{68565651900882}{49297184807}$ | 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: | \( 9655295.79813 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 9655295.79813 \cdot 1}{2\cdot\sqrt{1704986606307341449}}\cr\approx \mathstrut & 0.473243950900 \end{aligned}\]
Galois group
A cyclic group of order 7 |
The 7 conjugacy class representatives for $C_7$ |
Character table for $C_7$ |
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.1.0.1}{1} }^{7}$ | ${\href{/padicField/13.1.0.1}{1} }^{7}$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(1093\) | Deg $7$ | $7$ | $1$ | $6$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.1093.7t1.a.c | $1$ | $ 1093 $ | 7.7.1704986606307341449.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.1093.7t1.a.f | $1$ | $ 1093 $ | 7.7.1704986606307341449.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.1093.7t1.a.b | $1$ | $ 1093 $ | 7.7.1704986606307341449.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.1093.7t1.a.e | $1$ | $ 1093 $ | 7.7.1704986606307341449.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.1093.7t1.a.a | $1$ | $ 1093 $ | 7.7.1704986606307341449.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.1093.7t1.a.d | $1$ | $ 1093 $ | 7.7.1704986606307341449.1 | $C_7$ (as 7T1) | $0$ | $1$ |