Normalized defining polynomial
\( x^{7} - x^{6} - 162x^{5} + 201x^{4} + 7822x^{3} - 12322x^{2} - 107717x + 193369 \)
Invariants
Degree: | $7$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[7, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(2963706958323721\) \(\medspace = 379^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(162.28\) | 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: | $379^{6/7}\approx 162.2789466236963$ | ||
Ramified primes: | \(379\) | 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: | \(379\) | ||
Dirichlet character group: | $\lbrace$$\chi_{379}(1,·)$, $\chi_{379}(195,·)$, $\chi_{379}(86,·)$, $\chi_{379}(119,·)$, $\chi_{379}(138,·)$, $\chi_{379}(125,·)$, $\chi_{379}(94,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{11}a^{5}+\frac{1}{11}a^{4}+\frac{4}{11}a^{3}-\frac{1}{11}a^{2}-\frac{5}{11}a$, $\frac{1}{940774879}a^{6}-\frac{3102936}{85524989}a^{5}+\frac{320815113}{940774879}a^{4}-\frac{287829481}{940774879}a^{3}+\frac{124419951}{940774879}a^{2}+\frac{239015727}{940774879}a-\frac{14044702}{85524989}$
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: | $\frac{4153040}{940774879}a^{6}+\frac{342117}{85524989}a^{5}-\frac{655967001}{940774879}a^{4}-\frac{398642205}{940774879}a^{3}+\frac{30877906374}{940774879}a^{2}+\frac{7550380106}{940774879}a-\frac{38054266207}{85524989}$, $\frac{912806}{940774879}a^{6}-\frac{887709}{85524989}a^{5}-\frac{49598350}{940774879}a^{4}+\frac{716599259}{940774879}a^{3}-\frac{519525187}{940774879}a^{2}-\frac{8322887872}{940774879}a+\frac{1219422145}{85524989}$, $\frac{4160965}{940774879}a^{6}+\frac{1896154}{85524989}a^{5}-\frac{597544479}{940774879}a^{4}-\frac{2719472214}{940774879}a^{3}+\frac{22934595891}{940774879}a^{2}+\frac{79725650925}{940774879}a-\frac{17222421552}{85524989}$, $\frac{2045905}{940774879}a^{6}-\frac{2268583}{85524989}a^{5}-\frac{179118763}{940774879}a^{4}+\frac{2669135207}{940774879}a^{3}-\frac{216959583}{940774879}a^{2}-\frac{48685576458}{940774879}a+\frac{6888608096}{85524989}$, $\frac{7404820}{940774879}a^{6}+\frac{2268279}{85524989}a^{5}-\frac{1047264566}{940774879}a^{4}-\frac{3259073502}{940774879}a^{3}+\frac{38831004962}{940774879}a^{2}+\frac{94792252789}{940774879}a-\frac{25162461417}{85524989}$, $\frac{2907697}{940774879}a^{6}-\frac{1183829}{85524989}a^{5}-\frac{400046633}{940774879}a^{4}+\frac{1907216922}{940774879}a^{3}+\frac{12512041857}{940774879}a^{2}-\frac{68918103227}{940774879}a+\frac{4293000711}{85524989}$ | 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: | \( 269113.064107 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 269113.064107 \cdot 1}{2\cdot\sqrt{2963706958323721}}\cr\approx \mathstrut & 0.316371333606 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.1.0.1}{1} }^{7}$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(379\) | Deg $7$ | $7$ | $1$ | $6$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.379.7t1.a.a | $1$ | $ 379 $ | 7.7.2963706958323721.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.379.7t1.a.b | $1$ | $ 379 $ | 7.7.2963706958323721.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.379.7t1.a.c | $1$ | $ 379 $ | 7.7.2963706958323721.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.379.7t1.a.d | $1$ | $ 379 $ | 7.7.2963706958323721.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.379.7t1.a.e | $1$ | $ 379 $ | 7.7.2963706958323721.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.379.7t1.a.f | $1$ | $ 379 $ | 7.7.2963706958323721.1 | $C_7$ (as 7T1) | $0$ | $1$ |