Normalized defining polynomial
\( x^{7} - x^{6} - 354x^{5} - 979x^{4} + 30030x^{3} + 111552x^{2} - 715705x - 2921075 \)
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: | \(319913861015774089\) \(\medspace = 827^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(316.75\) | 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: | $827^{6/7}\approx 316.7517392217116$ | ||
Ramified primes: | \(827\) | 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: | \(827\) | ||
Dirichlet character group: | $\lbrace$$\chi_{827}(400,·)$, $\chi_{827}(1,·)$, $\chi_{827}(389,·)$, $\chi_{827}(807,·)$, $\chi_{827}(490,·)$, $\chi_{827}(124,·)$, $\chi_{827}(270,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{1}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{25}a^{5}+\frac{2}{25}a^{4}-\frac{8}{25}a^{3}+\frac{2}{25}a^{2}-\frac{9}{25}a+\frac{1}{5}$, $\frac{1}{110418475}a^{6}-\frac{128982}{22083695}a^{5}-\frac{2798577}{110418475}a^{4}+\frac{41936378}{110418475}a^{3}-\frac{36899353}{110418475}a^{2}+\frac{2151018}{110418475}a+\frac{6627193}{22083695}$
Monogenic: | No | |
Index: | $25$ | |
Inessential primes: | $5$ |
Class group and class number
$C_{2}\times C_{2}\times C_{2}$, which has order $8$
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: | $\frac{14506}{110418475}a^{6}-\frac{411258}{110418475}a^{5}-\frac{1909813}{110418475}a^{4}+\frac{82304622}{110418475}a^{3}+\frac{227838481}{110418475}a^{2}-\frac{589511768}{22083695}a-\frac{349941612}{4416739}$, $\frac{426652}{110418475}a^{6}-\frac{2551837}{110418475}a^{5}-\frac{112088158}{110418475}a^{4}+\frac{3357022}{110418475}a^{3}+\frac{1129066098}{22083695}a^{2}+\frac{12291386519}{110418475}a-\frac{3991140501}{22083695}$, $\frac{112208}{22083695}a^{6}-\frac{5464259}{110418475}a^{5}-\frac{170512313}{110418475}a^{4}+\frac{957625527}{110418475}a^{3}+\frac{14604104122}{110418475}a^{2}-\frac{44260683464}{110418475}a-\frac{68531070279}{22083695}$, $\frac{7134}{110418475}a^{6}+\frac{1454098}{110418475}a^{5}+\frac{16278918}{110418475}a^{4}-\frac{152698117}{110418475}a^{3}-\frac{1883870716}{110418475}a^{2}+\frac{782968376}{22083695}a+\frac{2006406975}{4416739}$, $\frac{26588}{22083695}a^{6}+\frac{3402068}{110418475}a^{5}-\frac{47400844}{110418475}a^{4}-\frac{1093229224}{110418475}a^{3}-\frac{1278255909}{110418475}a^{2}+\frac{40681566033}{110418475}a+\frac{26543460263}{22083695}$, $\frac{215253}{22083695}a^{6}-\frac{12360778}{110418475}a^{5}-\frac{253817761}{110418475}a^{4}+\frac{1616553644}{110418475}a^{3}+\frac{16061247774}{110418475}a^{2}-\frac{47147965078}{110418475}a-\frac{62525794673}{22083695}$ | 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: | \( 758444.158782 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 758444.158782 \cdot 8}{2\cdot\sqrt{319913861015774089}}\cr\approx \mathstrut & 0.686557701523 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.1.0.1}{1} }^{7}$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.1.0.1}{1} }^{7}$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(827\) | Deg $7$ | $7$ | $1$ | $6$ |