Normalized defining polynomial
\( x^{7} - x^{6} - 732x^{5} + 3104x^{4} + 155808x^{3} - 1028208x^{2} - 6517504x + 42320896 \)
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)
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Discriminant: | \(24914511498062181241\) \(\medspace = 1709^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(590.09\) | 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))
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Galois root discriminant: | $1709^{6/7}\approx 590.0947734443631$ | ||
Ramified primes: | \(1709\) | 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)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(1709\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1709}(880,·)$, $\chi_{1709}(1,·)$, $\chi_{1709}(866,·)$, $\chi_{1709}(1414,·)$, $\chi_{1709}(1575,·)$, $\chi_{1709}(168,·)$, $\chi_{1709}(223,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{3}+\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{128}a^{4}+\frac{3}{128}a^{3}-\frac{5}{32}a^{2}-\frac{7}{32}a+\frac{1}{4}$, $\frac{1}{512}a^{5}+\frac{1}{512}a^{4}-\frac{13}{256}a^{3}+\frac{3}{128}a^{2}-\frac{21}{64}a-\frac{1}{8}$, $\frac{1}{42131456}a^{6}+\frac{39453}{42131456}a^{5}-\frac{73227}{21065728}a^{4}-\frac{188133}{10532864}a^{3}-\frac{214847}{5266432}a^{2}+\frac{36991}{164576}a-\frac{1061}{2224}$
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)
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Fundamental units: | $\frac{24755323493}{5266432}a^{6}+\frac{99151455153}{5266432}a^{5}-\frac{8886153198695}{2633216}a^{4}-\frac{3740893682113}{1316608}a^{3}+\frac{480582088055989}{658304}a^{2}-\frac{40358075763353}{41144}a-\frac{10249837078365}{278}$, $\frac{407686303255}{21065728}a^{6}+\frac{5997064705147}{21065728}a^{5}-\frac{102172157935021}{10532864}a^{4}-\frac{486446108801843}{5266432}a^{3}+\frac{41\!\cdots\!47}{2633216}a^{2}+\frac{389655916361617}{82288}a-\frac{58140881423643}{1112}$, $\frac{2433394651}{142336}a^{6}+\frac{26104734347}{142336}a^{5}-\frac{738181620211}{71168}a^{4}-\frac{2437116017545}{35584}a^{3}+\frac{33165321924797}{17792}a^{2}+\frac{18880154714435}{4448}a-\frac{34375348179643}{556}$, $\frac{1875966459}{658304}a^{6}+\frac{51929985853}{1316608}a^{5}-\frac{1999065672373}{1316608}a^{4}-\frac{8924588269687}{658304}a^{3}+\frac{82143867364841}{329152}a^{2}+\frac{117369995716653}{164576}a-\frac{4605842543007}{556}$, $\frac{451398395}{2633216}a^{6}-\frac{2130659103}{2633216}a^{5}-\frac{48221470431}{329152}a^{4}+\frac{229723082473}{329152}a^{3}+\frac{6074103956553}{164576}a^{2}-\frac{24368739102451}{164576}a-\frac{1305316275731}{556}$, $\frac{14889174841}{10532864}a^{6}-\frac{240987664987}{10532864}a^{5}-\frac{2883066395147}{5266432}a^{4}+\frac{30441633891523}{2633216}a^{3}-\frac{24530942561679}{1316608}a^{2}-\frac{11013151316259}{20572}a+\frac{1209402629523}{556}$ (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: | \( 1611972567.68 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 1611972567.68 \cdot 1}{2\cdot\sqrt{24914511498062181241}}\cr\approx \mathstrut & 20.6686177118 \end{aligned}\] (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 | ${\href{/padicField/2.1.0.1}{1} }^{7}$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.1.0.1}{1} }^{7}$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(1709\) | Deg $7$ | $7$ | $1$ | $6$ |