Normalized defining polynomial
\( x^{7} - x^{6} - 630x^{5} - 1321x^{4} + 105050x^{3} + 532744x^{2} - 1423169x - 1017227 \)
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: | \(10131553366728232321\) \(\medspace = 1471^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(518.92\) | 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: | $1471^{6/7}\approx 518.9154203798864$ | ||
Ramified primes: | \(1471\) | 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: | \(1471\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1471}(1,·)$, $\chi_{1471}(1347,·)$, $\chi_{1471}(1217,·)$, $\chi_{1471}(666,·)$, $\chi_{1471}(785,·)$, $\chi_{1471}(605,·)$, $\chi_{1471}(1263,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{17}a^{5}-\frac{7}{17}a^{4}-\frac{6}{17}a^{3}-\frac{4}{17}a^{2}+\frac{7}{17}a+\frac{8}{17}$, $\frac{1}{187739448643}a^{6}-\frac{3924384658}{187739448643}a^{5}-\frac{415164067}{2376448717}a^{4}+\frac{2079950653}{187739448643}a^{3}+\frac{19525671711}{187739448643}a^{2}-\frac{84834159817}{187739448643}a+\frac{3345325815}{187739448643}$
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: | $\frac{25954750178}{187739448643}a^{6}-\frac{297547594546}{187739448643}a^{5}-\frac{167811872405}{2376448717}a^{4}+\frac{104293598920238}{187739448643}a^{3}+\frac{16\!\cdots\!18}{187739448643}a^{2}-\frac{33\!\cdots\!92}{187739448643}a-\frac{25\!\cdots\!21}{187739448643}$, $\frac{1956875816}{187739448643}a^{6}+\frac{2820254548}{187739448643}a^{5}-\frac{15387591228}{2376448717}a^{4}-\frac{5463457839360}{187739448643}a^{3}+\frac{188540181561700}{187739448643}a^{2}+\frac{14\!\cdots\!13}{187739448643}a+\frac{807828828188717}{187739448643}$, $\frac{25294836122}{187739448643}a^{6}-\frac{295906171157}{187739448643}a^{5}-\frac{161632416870}{2376448717}a^{4}+\frac{6070133683715}{11043496979}a^{3}+\frac{15\!\cdots\!40}{187739448643}a^{2}-\frac{31\!\cdots\!88}{187739448643}a-\frac{24\!\cdots\!00}{187739448643}$, $\frac{2273882885}{187739448643}a^{6}+\frac{934839425}{187739448643}a^{5}-\frac{18817449637}{2376448717}a^{4}-\frac{4420166218837}{187739448643}a^{3}+\frac{260155892819628}{187739448643}a^{2}+\frac{13\!\cdots\!60}{187739448643}a-\frac{46\!\cdots\!65}{187739448643}$, $\frac{126929473624}{187739448643}a^{6}-\frac{1485261422345}{187739448643}a^{5}-\frac{811117110750}{2376448717}a^{4}+\frac{518072398378472}{187739448643}a^{3}+\frac{77\!\cdots\!92}{187739448643}a^{2}-\frac{15\!\cdots\!03}{187739448643}a-\frac{12\!\cdots\!57}{187739448643}$, $\frac{1077035225}{187739448643}a^{6}-\frac{21173670583}{187739448643}a^{5}-\frac{3912584329}{2376448717}a^{4}+\frac{4471348727272}{187739448643}a^{3}+\frac{36033549744353}{187739448643}a^{2}-\frac{82118952826937}{187739448643}a-\frac{60850932870347}{187739448643}$ | 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: | \( 10351757.8241 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 10351757.8241 \cdot 1}{2\cdot\sqrt{10131553366728232321}}\cr\approx \mathstrut & 0.208140243869 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.1.0.1}{1} }^{7}$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(1471\) | Deg $7$ | $7$ | $1$ | $6$ |