Normalized defining polynomial
\( x^{7} - x^{6} - 180x^{5} + 103x^{4} + 6180x^{3} - 11596x^{2} - 25209x + 49213 \)
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: | \(5567914722008521\) \(\medspace = 421^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(177.58\) | 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: | $421^{6/7}\approx 177.57615723232306$ | ||
Ramified primes: | \(421\) | 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: | \(421\) | ||
Dirichlet character group: | $\lbrace$$\chi_{421}(385,·)$, $\chi_{421}(370,·)$, $\chi_{421}(33,·)$, $\chi_{421}(1,·)$, $\chi_{421}(152,·)$, $\chi_{421}(75,·)$, $\chi_{421}(247,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{13}a^{5}+\frac{3}{13}a^{4}-\frac{3}{13}a^{3}+\frac{1}{13}a^{2}-\frac{5}{13}a+\frac{2}{13}$, $\frac{1}{1690863421}a^{6}+\frac{32924542}{1690863421}a^{5}-\frac{3279054}{9773777}a^{4}+\frac{232299659}{1690863421}a^{3}-\frac{180289649}{1690863421}a^{2}-\frac{720887298}{1690863421}a+\frac{29412496}{1690863421}$
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: | $\frac{32963050}{1690863421}a^{6}-\frac{109104616}{1690863421}a^{5}-\frac{33100506}{9773777}a^{4}+\frac{16573478319}{1690863421}a^{3}+\frac{172799284325}{1690863421}a^{2}-\frac{770136086047}{1690863421}a+\frac{790902456800}{1690863421}$, $\frac{19857069}{1690863421}a^{6}+\frac{74181716}{1690863421}a^{5}-\frac{18672565}{9773777}a^{4}-\frac{13300535691}{1690863421}a^{3}+\frac{60761133486}{1690863421}a^{2}+\frac{64685656076}{1690863421}a-\frac{199375988076}{1690863421}$, $\frac{717215}{130066417}a^{6}-\frac{2830398}{130066417}a^{5}-\frac{713984}{751829}a^{4}+\frac{443185933}{130066417}a^{3}+\frac{277704791}{10005109}a^{2}-\frac{19627111775}{130066417}a+\frac{24653933237}{130066417}$, $\frac{56424}{1690863421}a^{6}-\frac{4276203}{1690863421}a^{5}-\frac{496115}{9773777}a^{4}-\frac{167213759}{1690863421}a^{3}+\frac{1782314649}{1690863421}a^{2}+\frac{845951726}{1690863421}a-\frac{6580127466}{1690863421}$, $\frac{839}{9773777}a^{6}-\frac{10380}{9773777}a^{5}-\frac{254317}{9773777}a^{4}-\frac{225085}{9773777}a^{3}+\frac{2723802}{9773777}a^{2}+\frac{1688095}{9773777}a-\frac{7717413}{9773777}$, $\frac{636606}{1690863421}a^{6}+\frac{18017736}{1690863421}a^{5}-\frac{1706618}{9773777}a^{4}-\frac{1248946727}{1690863421}a^{3}+\frac{12481906712}{1690863421}a^{2}+\frac{1134453285}{1690863421}a-\frac{59630315313}{1690863421}$ | 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: | \( 428479.510505 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 428479.510505 \cdot 1}{2\cdot\sqrt{5567914722008521}}\cr\approx \mathstrut & 0.367505417625 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.1.0.1}{1} }^{7}$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.1.0.1}{1} }^{7}$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(421\) | Deg $7$ | $7$ | $1$ | $6$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.421.7t1.a.a | $1$ | $ 421 $ | 7.7.5567914722008521.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.421.7t1.a.b | $1$ | $ 421 $ | 7.7.5567914722008521.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.421.7t1.a.c | $1$ | $ 421 $ | 7.7.5567914722008521.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.421.7t1.a.d | $1$ | $ 421 $ | 7.7.5567914722008521.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.421.7t1.a.e | $1$ | $ 421 $ | 7.7.5567914722008521.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.421.7t1.a.f | $1$ | $ 421 $ | 7.7.5567914722008521.1 | $C_7$ (as 7T1) | $0$ | $1$ |