Normalized defining polynomial
\( x^{7} - x^{6} - 408x^{5} - 992x^{4} + 48064x^{3} + 204560x^{2} - 1603520x - 8290816 \)
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)
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Discriminant: | \(749130369924173329\) \(\medspace = 953^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(357.69\) | 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: | $953^{6/7}\approx 357.6911546007666$ | ||
Ramified primes: | \(953\) | 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: | \(953\) | ||
Dirichlet character group: | $\lbrace$$\chi_{953}(528,·)$, $\chi_{953}(1,·)$, $\chi_{953}(754,·)$, $\chi_{953}(711,·)$, $\chi_{953}(431,·)$, $\chi_{953}(508,·)$, $\chi_{953}(879,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{3}+\frac{1}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{16}a^{4}-\frac{1}{16}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a$, $\frac{1}{128}a^{5}+\frac{3}{128}a^{4}+\frac{1}{32}a^{2}+\frac{1}{8}a$, $\frac{1}{4507648}a^{6}+\frac{16189}{4507648}a^{5}+\frac{46231}{2253824}a^{4}-\frac{35951}{1126912}a^{3}-\frac{133177}{563456}a^{2}+\frac{59013}{140864}a+\frac{3839}{17608}$
Monogenic: | No | |
Index: | $4096$ | |
Inessential primes: | $2$ |
Class group and class number
$C_{71}$, which has order $71$
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: | $\frac{150337}{2253824}a^{6}-\frac{1275059}{2253824}a^{5}-\frac{25936497}{1126912}a^{4}+\frac{59771665}{563456}a^{3}+\frac{682848111}{281728}a^{2}-\frac{314309043}{70432}a-\frac{655653965}{8804}$, $\frac{2809}{2253824}a^{6}-\frac{24171}{2253824}a^{5}-\frac{365889}{1126912}a^{4}+\frac{858313}{563456}a^{3}+\frac{7365519}{281728}a^{2}-\frac{3656659}{70432}a-\frac{5882221}{8804}$, $\frac{12853}{281728}a^{6}+\frac{131139}{281728}a^{5}-\frac{952597}{70432}a^{4}-\frac{13848955}{70432}a^{3}+\frac{225683}{8804}a^{2}+\frac{42948551}{4402}a+\frac{75960009}{2201}$, $\frac{17387}{2253824}a^{6}-\frac{179425}{2253824}a^{5}-\frac{2663219}{1126912}a^{4}+\frac{7681051}{563456}a^{3}+\frac{66849117}{281728}a^{2}-\frac{36759849}{70432}a-\frac{64069943}{8804}$, $\frac{5565}{2253824}a^{6}-\frac{96391}{2253824}a^{5}-\frac{840157}{1126912}a^{4}+\frac{5453261}{563456}a^{3}+\frac{25927299}{281728}a^{2}-\frac{29158311}{70432}a-\frac{29056473}{8804}$, $\frac{21463}{2253824}a^{6}-\frac{258277}{2253824}a^{5}-\frac{2912991}{1126912}a^{4}+\frac{10601479}{563456}a^{3}+\frac{67050385}{281728}a^{2}-\frac{47911613}{70432}a-\frac{59849771}{8804}$ | 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: | \( 4478301.42813 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 4478301.42813 \cdot 71}{2\cdot\sqrt{749130369924173329}}\cr\approx \mathstrut & 23.5110996796 \end{aligned}\]
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 | ${\href{/padicField/2.1.0.1}{1} }^{7}$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.1.0.1}{1} }^{7}$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(953\) | Deg $7$ | $7$ | $1$ | $6$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.953.7t1.a.a | $1$ | $ 953 $ | 7.7.749130369924173329.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.953.7t1.a.f | $1$ | $ 953 $ | 7.7.749130369924173329.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.953.7t1.a.b | $1$ | $ 953 $ | 7.7.749130369924173329.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.953.7t1.a.d | $1$ | $ 953 $ | 7.7.749130369924173329.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.953.7t1.a.c | $1$ | $ 953 $ | 7.7.749130369924173329.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.953.7t1.a.e | $1$ | $ 953 $ | 7.7.749130369924173329.1 | $C_7$ (as 7T1) | $0$ | $1$ |