Normalized defining polynomial
\( x^{7} - x^{6} - 588x^{5} + 1513x^{4} + 65884x^{3} + 33346x^{2} - 962361x - 55903 \)
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: | \(6699204317182001689\) \(\medspace = 1373^{6}\) | 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: | \(489.14\) | 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: | $1373^{6/7}\approx 489.1385469162994$ | ||
Ramified primes: | \(1373\) | 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: | \(1373\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1373}(1,·)$, $\chi_{1373}(628,·)$, $\chi_{1373}(1105,·)$, $\chi_{1373}(1049,·)$, $\chi_{1373}(428,·)$, $\chi_{1373}(333,·)$, $\chi_{1373}(575,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{139849056302161}a^{6}+\frac{946855832682}{139849056302161}a^{5}+\frac{17298284062213}{139849056302161}a^{4}-\frac{61225951001361}{139849056302161}a^{3}-\frac{69755962180421}{139849056302161}a^{2}-\frac{15332329090918}{139849056302161}a+\frac{36947496246697}{139849056302161}$
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)
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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: | $\frac{19270406267}{139849056302161}a^{6}+\frac{1601199609134}{139849056302161}a^{5}+\frac{9552310884301}{139849056302161}a^{4}-\frac{514960206238261}{139849056302161}a^{3}-\frac{24\!\cdots\!05}{139849056302161}a^{2}+\frac{14\!\cdots\!90}{139849056302161}a+\frac{847601202174892}{139849056302161}$, $\frac{1065840467220}{139849056302161}a^{6}-\frac{981352531387}{139849056302161}a^{5}-\frac{626236515866334}{139849056302161}a^{4}+\frac{15\!\cdots\!47}{139849056302161}a^{3}+\frac{70\!\cdots\!46}{139849056302161}a^{2}+\frac{40\!\cdots\!85}{139849056302161}a-\frac{98\!\cdots\!19}{139849056302161}$, $\frac{341181179550}{139849056302161}a^{6}-\frac{3074627520778}{139849056302161}a^{5}-\frac{172793347899885}{139849056302161}a^{4}+\frac{19\!\cdots\!90}{139849056302161}a^{3}+\frac{58\!\cdots\!90}{139849056302161}a^{2}-\frac{38\!\cdots\!27}{139849056302161}a-\frac{903687205908154}{139849056302161}$, $\frac{21211075905}{139849056302161}a^{6}-\frac{914924751082}{139849056302161}a^{5}+\frac{1404475723811}{139849056302161}a^{4}+\frac{122236715205427}{139849056302161}a^{3}+\frac{86483074627023}{139849056302161}a^{2}-\frac{18\!\cdots\!43}{139849056302161}a-\frac{98088389977282}{139849056302161}$, $\frac{836992138282}{139849056302161}a^{6}-\frac{5047429253533}{139849056302161}a^{5}-\frac{466857540500485}{139849056302161}a^{4}+\frac{36\!\cdots\!24}{139849056302161}a^{3}+\frac{36\!\cdots\!89}{139849056302161}a^{2}-\frac{15\!\cdots\!93}{139849056302161}a-\frac{93\!\cdots\!83}{139849056302161}$, $\frac{569784557261}{139849056302161}a^{6}+\frac{9502543323970}{139849056302161}a^{5}-\frac{166816090892206}{139849056302161}a^{4}-\frac{20\!\cdots\!21}{139849056302161}a^{3}+\frac{735240808743657}{139849056302161}a^{2}+\frac{31\!\cdots\!05}{139849056302161}a+\frac{18\!\cdots\!19}{139849056302161}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 5167793.74822 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 5167793.74822 \cdot 1}{2\cdot\sqrt{6699204317182001689}}\cr\approx \mathstrut & 0.127783133729 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(1373\) | Deg $7$ | $7$ | $1$ | $6$ |