Normalized defining polynomial
\( x^{7} - 609x^{5} - 6293x^{4} + 2639x^{3} + 125454x^{2} - 133168x + 30479 \)
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: | \(8233120419813614521\) \(\medspace = 7^{12}\cdot 29^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(503.76\) | 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: | $7^{12/7}29^{6/7}\approx 503.75978549322235$ | ||
Ramified primes: | \(7\), \(29\) | 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: | \(1421=7^{2}\cdot 29\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1421}(1184,·)$, $\chi_{1421}(1,·)$, $\chi_{1421}(36,·)$, $\chi_{1421}(1205,·)$, $\chi_{1421}(1415,·)$, $\chi_{1421}(1296,·)$, $\chi_{1421}(750,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{89}a^{5}+\frac{16}{89}a^{4}+\frac{38}{89}a^{3}+\frac{37}{89}a^{2}+\frac{22}{89}a+\frac{9}{89}$, $\frac{1}{5275929523}a^{6}-\frac{28521658}{5275929523}a^{5}-\frac{342624504}{5275929523}a^{4}+\frac{346424581}{5275929523}a^{3}-\frac{1600645233}{5275929523}a^{2}-\frac{564548390}{5275929523}a-\frac{2420916076}{5275929523}$
Monogenic: | No | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{7}$, which has order $7$
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: | $\frac{4305836}{5275929523}a^{6}-\frac{51166114}{5275929523}a^{5}-\frac{2004614436}{5275929523}a^{4}-\frac{3554500390}{5275929523}a^{3}+\frac{52753855613}{5275929523}a^{2}-\frac{57055282076}{5275929523}a+\frac{13360182285}{5275929523}$, $\frac{12687684}{5275929523}a^{6}+\frac{28758539}{5275929523}a^{5}-\frac{7469249569}{5275929523}a^{4}-\frac{97688374677}{5275929523}a^{3}-\frac{295869059565}{5275929523}a^{2}+\frac{201097257443}{5275929523}a-\frac{30435438467}{5275929523}$, $\frac{5143373}{5275929523}a^{6}-\frac{68164991}{5275929523}a^{5}-\frac{2229460299}{5275929523}a^{4}-\frac{2899268152}{5275929523}a^{3}+\frac{54212476001}{5275929523}a^{2}-\frac{54088898435}{5275929523}a+\frac{12089245143}{5275929523}$, $\frac{17536}{5275929523}a^{6}-\frac{9531929}{5275929523}a^{5}-\frac{224457694}{5275929523}a^{4}-\frac{1309515084}{5275929523}a^{3}+\frac{1757141394}{5275929523}a^{2}+\frac{21731661443}{5275929523}a-\frac{7382814489}{5275929523}$, $\frac{747343}{5275929523}a^{6}+\frac{5179510}{5275929523}a^{5}-\frac{507231615}{5275929523}a^{4}-\frac{7051762719}{5275929523}a^{3}-\frac{9017037190}{5275929523}a^{2}+\frac{110137276679}{5275929523}a-\frac{74012073059}{5275929523}$, $\frac{7634196}{5275929523}a^{6}-\frac{4078585}{5275929523}a^{5}-\frac{4427181921}{5275929523}a^{4}-\frac{46638400881}{5275929523}a^{3}-\frac{79044363427}{5275929523}a^{2}+\frac{126836362250}{5275929523}a-\frac{29826898833}{5275929523}$ | 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: | \( 934548.225652 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 934548.225652 \cdot 7}{2\cdot\sqrt{8233120419813614521}}\cr\approx \mathstrut & 0.145914179537 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | R | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | R | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
In the table, R denotes a ramified prime. 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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.7.12.5 | $x^{7} + 42 x^{6} + 203$ | $7$ | $1$ | $12$ | $C_7$ | $[2]$ |
\(29\) | 29.7.6.3 | $x^{7} + 87$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |