Normalized defining polynomial
\( x^{7} - x^{6} - 738x^{5} + 9283x^{4} - 11418x^{3} - 331708x^{2} + 1727391x - 2487539 \)
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: | \(26164453299826084489\) \(\medspace = 1723^{6}\) | 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: | \(594.24\) | 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: | $1723^{6/7}\approx 594.2357955136594$ | ||
Ramified primes: | \(1723\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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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: | \(1723\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1723}(1,·)$, $\chi_{1723}(1515,·)$, $\chi_{1723}(189,·)$, $\chi_{1723}(1331,·)$, $\chi_{1723}(555,·)$, $\chi_{1723}(1261,·)$, $\chi_{1723}(317,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{520230244711}a^{6}+\frac{247223450017}{520230244711}a^{5}-\frac{164597208124}{520230244711}a^{4}-\frac{239484666501}{520230244711}a^{3}+\frac{152312436061}{520230244711}a^{2}+\frac{37765870018}{520230244711}a-\frac{31075913472}{520230244711}$
Monogenic: | No | |
Index: | $1$ | |
Inessential primes: | None |
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: | $\frac{323542124}{520230244711}a^{6}+\frac{145066048}{520230244711}a^{5}-\frac{224219488183}{520230244711}a^{4}+\frac{2669416477059}{520230244711}a^{3}-\frac{9640303338800}{520230244711}a^{2}+\frac{11118304572295}{520230244711}a-\frac{22774910116608}{520230244711}$, $\frac{842375159}{520230244711}a^{6}+\frac{1951759184}{520230244711}a^{5}-\frac{614952599983}{520230244711}a^{4}+\frac{5819356249960}{520230244711}a^{3}+\frac{8851843682838}{520230244711}a^{2}-\frac{250434756849548}{520230244711}a+\frac{655471177515677}{520230244711}$, $\frac{15027568250}{520230244711}a^{6}+\frac{45368411804}{520230244711}a^{5}-\frac{10535988338710}{520230244711}a^{4}+\frac{86897209988226}{520230244711}a^{3}+\frac{235136816781469}{520230244711}a^{2}-\frac{37\!\cdots\!39}{520230244711}a+\frac{78\!\cdots\!03}{520230244711}$, $\frac{44101594892}{520230244711}a^{6}+\frac{227334308011}{520230244711}a^{5}-\frac{31147441916639}{520230244711}a^{4}+\frac{217694075286084}{520230244711}a^{3}+\frac{836065408930390}{520230244711}a^{2}-\frac{94\!\cdots\!36}{520230244711}a+\frac{17\!\cdots\!49}{520230244711}$, $\frac{236405338}{520230244711}a^{6}+\frac{3949905988}{520230244711}a^{5}-\frac{128143367178}{520230244711}a^{4}-\frac{331104558303}{520230244711}a^{3}+\frac{6540429233627}{520230244711}a^{2}+\frac{5390286714320}{520230244711}a-\frac{62219261962151}{520230244711}$, $\frac{17483135448}{520230244711}a^{6}+\frac{112426325738}{520230244711}a^{5}-\frac{12063416574581}{520230244711}a^{4}+\frac{72700732365958}{520230244711}a^{3}+\frac{338251671065575}{520230244711}a^{2}-\frac{32\!\cdots\!02}{520230244711}a+\frac{58\!\cdots\!62}{520230244711}$ (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: | \( 13885411.1162 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 13885411.1162 \cdot 1}{2\cdot\sqrt{26164453299826084489}}\cr\approx \mathstrut & 0.173733226160 \end{aligned}\] (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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.1.0.1}{1} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(1723\) | Deg $7$ | $7$ | $1$ | $6$ |