Normalized defining polynomial
\( x^{7} - x^{6} - 612x^{5} + 1983x^{4} + 118732x^{3} - 607744x^{2} - 6889121x + 43521943 \)
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: | \(8515170979269874921\) \(\medspace = 1429^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(506.19\) | 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: | $1429^{6/7}\approx 506.1897409342453$ | ||
Ramified primes: | \(1429\) | 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)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(1429\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1429}(1,·)$, $\chi_{1429}(1362,·)$, $\chi_{1429}(756,·)$, $\chi_{1429}(1365,·)$, $\chi_{1429}(1238,·)$, $\chi_{1429}(792,·)$, $\chi_{1429}(202,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{31}a^{5}+\frac{7}{31}a^{4}-\frac{14}{31}a^{3}-\frac{8}{31}a^{2}+\frac{7}{31}a+\frac{9}{31}$, $\frac{1}{55029250007}a^{6}-\frac{66998871}{55029250007}a^{5}+\frac{2666704154}{55029250007}a^{4}-\frac{2511326754}{55029250007}a^{3}-\frac{4371266313}{55029250007}a^{2}-\frac{2350045973}{55029250007}a+\frac{19131882898}{55029250007}$
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)
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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: | $\frac{36600990}{55029250007}a^{6}+\frac{857289323}{55029250007}a^{5}-\frac{16009435526}{55029250007}a^{4}-\frac{267238379663}{55029250007}a^{3}+\frac{2844354474953}{55029250007}a^{2}+\frac{18532139098248}{55029250007}a-\frac{167153063339974}{55029250007}$, $\frac{713055355}{55029250007}a^{6}+\frac{5918637395}{55029250007}a^{5}-\frac{384169235197}{55029250007}a^{4}-\frac{2153227333131}{55029250007}a^{3}+\frac{65980980478632}{55029250007}a^{2}+\frac{174541225797585}{55029250007}a-\frac{34\!\cdots\!33}{55029250007}$, $\frac{205824673}{55029250007}a^{6}+\frac{2238092236}{55029250007}a^{5}-\frac{118953302064}{55029250007}a^{4}-\frac{834234901746}{55029250007}a^{3}+\frac{22508269552397}{55029250007}a^{2}+\frac{65459743071075}{55029250007}a-\frac{12\!\cdots\!43}{55029250007}$, $\frac{190055784}{55029250007}a^{6}+\frac{2302817647}{55029250007}a^{5}-\frac{87055419483}{55029250007}a^{4}-\frac{769821229633}{55029250007}a^{3}+\frac{12906719435314}{55029250007}a^{2}+\frac{54717781038176}{55029250007}a-\frac{639749044323232}{55029250007}$, $\frac{11204914}{1775137097}a^{6}+\frac{56854171}{1775137097}a^{5}-\frac{5009469218}{1775137097}a^{4}-\frac{14729559844}{1775137097}a^{3}+\frac{638030061333}{1775137097}a^{2}+\frac{650906149207}{1775137097}a-\frac{23175708938532}{1775137097}$, $\frac{22554475}{55029250007}a^{6}-\frac{1629297438}{55029250007}a^{5}-\frac{6018055290}{55029250007}a^{4}+\frac{612668418131}{55029250007}a^{3}-\frac{1643820674046}{55029250007}a^{2}-\frac{55432455587390}{55029250007}a+\frac{308856626340210}{55029250007}$ | 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: | \( 9587300.65259 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 9587300.65259 \cdot 1}{2\cdot\sqrt{8515170979269874921}}\cr\approx \mathstrut & 0.210271123136 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.1.0.1}{1} }^{7}$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.1.0.1}{1} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.1.0.1}{1} }^{7}$ |
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 |
---|---|---|---|---|---|---|---|
\(1429\) | Deg $7$ | $7$ | $1$ | $6$ |