Normalized defining polynomial
\( x^{7} - x^{6} - 144x^{5} - 399x^{4} + 2416x^{3} + 10808x^{2} + 10831x + 1237 \)
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: | \(1464803622199009\) \(\medspace = 337^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(146.74\) | 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: | $337^{6/7}\approx 146.73709434855056$ | ||
Ramified primes: | \(337\) | 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: | \(337\) | ||
Dirichlet character group: | $\lbrace$$\chi_{337}(64,·)$, $\chi_{337}(1,·)$, $\chi_{337}(52,·)$, $\chi_{337}(295,·)$, $\chi_{337}(8,·)$, $\chi_{337}(175,·)$, $\chi_{337}(79,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{328261781}a^{6}+\frac{23635027}{328261781}a^{5}-\frac{36964423}{328261781}a^{4}+\frac{98789236}{328261781}a^{3}+\frac{514975}{5563759}a^{2}+\frac{144155478}{328261781}a+\frac{118429002}{328261781}$
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: | $\frac{769869}{328261781}a^{6}-\frac{4181148}{328261781}a^{5}-\frac{93052135}{328261781}a^{4}+\frac{126551975}{328261781}a^{3}+\frac{23204489}{5563759}a^{2}+\frac{121201216}{328261781}a-\frac{1861902698}{328261781}$, $\frac{1157227}{328261781}a^{6}-\frac{8464572}{328261781}a^{5}-\frac{107391130}{328261781}a^{4}+\frac{138572169}{328261781}a^{3}+\frac{31002871}{5563759}a^{2}+\frac{2241638459}{328261781}a+\frac{273391735}{328261781}$, $\frac{4957545}{328261781}a^{6}-\frac{19752111}{328261781}a^{5}-\frac{651178285}{328261781}a^{4}-\frac{50748016}{328261781}a^{3}+\frac{197758164}{5563759}a^{2}+\frac{18897996832}{328261781}a+\frac{5483097883}{328261781}$, $\frac{174582}{328261781}a^{6}-\frac{303456}{328261781}a^{5}-\frac{24543507}{328261781}a^{4}-\frac{51574388}{328261781}a^{3}+\frac{6147528}{5563759}a^{2}+\frac{2075266955}{328261781}a+\frac{332012660}{328261781}$, $\frac{2797548}{328261781}a^{6}-\frac{6724129}{328261781}a^{5}-\frac{393122403}{328261781}a^{4}-\frac{559495506}{328261781}a^{3}+\frac{126619815}{5563759}a^{2}+\frac{19164706845}{328261781}a+\frac{2439097635}{328261781}$, $\frac{1139005}{328261781}a^{6}-\frac{6469894}{328261781}a^{5}-\frac{134849836}{328261781}a^{4}+\frac{188720781}{328261781}a^{3}+\frac{33189854}{5563759}a^{2}+\frac{2191289905}{328261781}a+\frac{253065585}{328261781}$ | 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: | \( 85721.5666782 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 85721.5666782 \cdot 1}{2\cdot\sqrt{1464803622199009}}\cr\approx \mathstrut & 0.143344281176 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.1.0.1}{1} }^{7}$ |
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 |
---|---|---|---|---|---|---|---|
\(337\) | Deg $7$ | $7$ | $1$ | $6$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.337.7t1.a.a | $1$ | $ 337 $ | 7.7.1464803622199009.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.337.7t1.a.b | $1$ | $ 337 $ | 7.7.1464803622199009.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.337.7t1.a.c | $1$ | $ 337 $ | 7.7.1464803622199009.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.337.7t1.a.d | $1$ | $ 337 $ | 7.7.1464803622199009.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.337.7t1.a.e | $1$ | $ 337 $ | 7.7.1464803622199009.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.337.7t1.a.f | $1$ | $ 337 $ | 7.7.1464803622199009.1 | $C_7$ (as 7T1) | $0$ | $1$ |