Normalized defining polynomial
\( x^{7} - x^{6} - 498x^{5} + 1780x^{4} + 72184x^{3} - 361600x^{2} - 2988416x + 16876544 \)
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)
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Discriminant: | \(2474447753486836009\) \(\medspace = 1163^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(424.27\) | 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: | $1163^{6/7}\approx 424.26749075235574$ | ||
Ramified primes: | \(1163\) | 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: | \(1163\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1163}(1,·)$, $\chi_{1163}(498,·)$, $\chi_{1163}(773,·)$, $\chi_{1163}(44,·)$, $\chi_{1163}(978,·)$, $\chi_{1163}(910,·)$, $\chi_{1163}(285,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{32}a^{4}-\frac{1}{32}a^{3}-\frac{3}{16}a^{2}-\frac{1}{4}a$, $\frac{1}{128}a^{5}+\frac{1}{128}a^{4}-\frac{1}{16}a^{3}-\frac{5}{32}a^{2}-\frac{1}{8}a$, $\frac{1}{42041344}a^{6}+\frac{52945}{42041344}a^{5}+\frac{1955}{328448}a^{4}+\frac{396797}{10510336}a^{3}-\frac{84095}{1313792}a^{2}-\frac{155209}{656896}a-\frac{17209}{82112}$
Monogenic: | No | |
Index: | $4096$ | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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: | $\frac{621676787}{21020672}a^{6}-\frac{8808799645}{21020672}a^{5}-\frac{1960101987}{164224}a^{4}+\frac{1148459518247}{5255168}a^{3}+\frac{297292895411}{656896}a^{2}-\frac{7776519484363}{328448}a+\frac{4296461709509}{41056}$, $\frac{26023969271}{21020672}a^{6}-\frac{505712875865}{21020672}a^{5}-\frac{28424566495}{164224}a^{4}+\frac{28346692235035}{5255168}a^{3}-\frac{6609267435657}{656896}a^{2}-\frac{86122556242031}{328448}a+\frac{46537270189409}{41056}$, $\frac{291188043}{10510336}a^{6}-\frac{3968248901}{10510336}a^{5}-\frac{185928965}{20528}a^{4}+\frac{431709803935}{2627584}a^{3}-\frac{23228250373}{328448}a^{2}-\frac{1531398774531}{164224}a+\frac{752714593373}{20528}$, $\frac{184340889}{2627584}a^{6}+\frac{1293897961}{2627584}a^{5}-\frac{142016745}{5132}a^{4}-\frac{84709707179}{656896}a^{3}+\frac{267143093073}{82112}a^{2}+\frac{310105514031}{41056}a-\frac{559574247189}{5132}$, $\frac{2110630997}{21020672}a^{6}+\frac{27481164901}{21020672}a^{5}-\frac{5343403015}{164224}a^{4}-\frac{1459486168831}{5255168}a^{3}+\frac{2389323575861}{656896}a^{2}+\frac{4426337214851}{328448}a-\frac{5213738606989}{41056}$, $\frac{29975532893}{21020672}a^{6}+\frac{284950898669}{21020672}a^{5}-\frac{93242286767}{164224}a^{4}-\frac{18013910398743}{5255168}a^{3}+\frac{43962592480253}{656896}a^{2}+\frac{61626872135867}{328448}a-\frac{94025410590293}{41056}$ (assuming GRH) | 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: | \( 458862243.387 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 458862243.387 \cdot 1}{2\cdot\sqrt{2474447753486836009}}\cr\approx \mathstrut & 18.6690901937 \end{aligned}\] (assuming GRH)
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 | ${\href{/padicField/2.1.0.1}{1} }^{7}$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.1.0.1}{1} }^{7}$ |
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 |
---|---|---|---|---|---|---|---|
\(1163\) | Deg $7$ | $7$ | $1$ | $6$ |