Normalized defining polynomial
\( x^{7} - x^{6} - 192x^{5} - 275x^{4} + 3952x^{3} - 4136x^{2} - 81x + 863 \)
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: | \(8193662024284801\) \(\medspace = 449^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(187.65\) | 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: | $449^{6/7}\approx 187.6523495907157$ | ||
Ramified primes: | \(449\) | 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: | \(449\) | ||
Dirichlet character group: | $\lbrace$$\chi_{449}(176,·)$, $\chi_{449}(1,·)$, $\chi_{449}(18,·)$, $\chi_{449}(324,·)$, $\chi_{449}(359,·)$, $\chi_{449}(25,·)$, $\chi_{449}(444,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{61883671}a^{6}+\frac{6661371}{61883671}a^{5}-\frac{1680743}{61883671}a^{4}+\frac{1281320}{61883671}a^{3}-\frac{18031354}{61883671}a^{2}-\frac{12248677}{61883671}a+\frac{7371865}{61883671}$
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: | $\frac{53473}{61883671}a^{6}+\frac{1081207}{61883671}a^{5}-\frac{19280147}{61883671}a^{4}-\frac{174850450}{61883671}a^{3}+\frac{1009024145}{61883671}a^{2}-\frac{429917054}{61883671}a-\frac{250781809}{61883671}$, $\frac{4849427}{61883671}a^{6}-\frac{830622}{61883671}a^{5}-\frac{932315587}{61883671}a^{4}-\frac{2113762613}{61883671}a^{3}+\frac{17405267735}{61883671}a^{2}-\frac{5530995448}{61883671}a-\frac{5083654973}{61883671}$, $\frac{155142}{61883671}a^{6}+\frac{1113982}{61883671}a^{5}-\frac{37924583}{61883671}a^{4}-\frac{231338496}{61883671}a^{3}+\frac{1206815036}{61883671}a^{2}-\frac{393663763}{61883671}a-\frac{357545947}{61883671}$, $\frac{10140031}{61883671}a^{6}-\frac{13523367}{61883671}a^{5}-\frac{1941523434}{61883671}a^{4}-\frac{2141901357}{61883671}a^{3}+\frac{40631076594}{61883671}a^{2}-\frac{55708771454}{61883671}a+\frac{20466062899}{61883671}$, $\frac{1179190}{61883671}a^{6}+\frac{3942118}{61883671}a^{5}-\frac{214541737}{61883671}a^{4}-\frac{1269653756}{61883671}a^{3}+\frac{94428288}{61883671}a^{2}+\frac{626449138}{61883671}a+\frac{153991322}{61883671}$, $\frac{450608}{61883671}a^{6}-\frac{398287}{61883671}a^{5}-\frac{85759717}{61883671}a^{4}-\frac{125375212}{61883671}a^{3}+\frac{1738847172}{61883671}a^{2}-\frac{1865622927}{61883671}a+\frac{462837679}{61883671}$ | 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: | \( 263186.113809 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 263186.113809 \cdot 1}{2\cdot\sqrt{8193662024284801}}\cr\approx \mathstrut & 0.186081810251 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.1.0.1}{1} }^{7}$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(449\) | Deg $7$ | $7$ | $1$ | $6$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.449.7t1.a.a | $1$ | $ 449 $ | 7.7.8193662024284801.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.449.7t1.a.b | $1$ | $ 449 $ | 7.7.8193662024284801.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.449.7t1.a.c | $1$ | $ 449 $ | 7.7.8193662024284801.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.449.7t1.a.d | $1$ | $ 449 $ | 7.7.8193662024284801.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.449.7t1.a.e | $1$ | $ 449 $ | 7.7.8193662024284801.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.449.7t1.a.f | $1$ | $ 449 $ | 7.7.8193662024284801.1 | $C_7$ (as 7T1) | $0$ | $1$ |