Normalized defining polynomial
\( x^{7} - x^{6} - 300x^{5} - 1631x^{4} + 5140x^{3} + 23794x^{2} - 59049x + 18773 \)
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: | \(118661028367354201\) \(\medspace = 701^{6}\) | 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: | \(274.91\) | 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: | $701^{6/7}\approx 274.90767987111883$ | ||
Ramified primes: | \(701\) | 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: | \(701\) | ||
Dirichlet character group: | $\lbrace$$\chi_{701}(1,·)$, $\chi_{701}(19,·)$, $\chi_{701}(550,·)$, $\chi_{701}(167,·)$, $\chi_{701}(361,·)$, $\chi_{701}(636,·)$, $\chi_{701}(369,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{1157}a^{5}-\frac{220}{1157}a^{4}+\frac{490}{1157}a^{3}-\frac{110}{1157}a^{2}+\frac{15}{89}a+\frac{216}{1157}$, $\frac{1}{102973}a^{6}+\frac{23}{102973}a^{5}+\frac{252}{102973}a^{4}+\frac{4417}{102973}a^{3}+\frac{8175}{102973}a^{2}+\frac{14048}{102973}a-\frac{30816}{102973}$
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: | $\frac{22088}{102973}a^{6}-\frac{166952}{102973}a^{5}-\frac{5547254}{102973}a^{4}+\frac{470193}{102973}a^{3}+\frac{114464561}{102973}a^{2}-\frac{224158472}{102973}a+\frac{82484047}{102973}$, $\frac{248}{102973}a^{6}-\frac{1683}{102973}a^{5}-\frac{62905}{102973}a^{4}-\frac{52862}{102973}a^{3}+\frac{1192402}{102973}a^{2}-\frac{1354670}{102973}a-\frac{279309}{102973}$, $\frac{1344}{102973}a^{6}-\frac{12876}{102973}a^{5}-\frac{325252}{102973}a^{4}+\frac{750062}{102973}a^{3}+\frac{9110635}{102973}a^{2}-\frac{21476805}{102973}a+\frac{6789968}{102973}$, $\frac{127}{102973}a^{6}+\frac{1230}{102973}a^{5}-\frac{7868}{102973}a^{4}-\frac{4745}{7921}a^{3}+\frac{91532}{102973}a^{2}+\frac{321648}{102973}a-\frac{56995}{102973}$, $\frac{7867}{102973}a^{6}-\frac{93179}{102973}a^{5}-\frac{1348430}{102973}a^{4}+\frac{1755171}{102973}a^{3}+\frac{21561041}{102973}a^{2}-\frac{44572135}{102973}a+\frac{13766907}{102973}$, $\frac{25265}{102973}a^{6}-\frac{183771}{102973}a^{5}-\frac{6389234}{102973}a^{4}-\frac{102192}{7921}a^{3}+\frac{128391187}{102973}a^{2}-\frac{223623560}{102973}a+\frac{66696753}{102973}$ | 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: | \( 1357576.32866 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 1357576.32866 \cdot 1}{2\cdot\sqrt{118661028367354201}}\cr\approx \mathstrut & 0.252226188368 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.1.0.1}{1} }^{7}$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(701\) | Deg $7$ | $7$ | $1$ | $6$ |