Normalized defining polynomial
\( x^{7} - x^{6} - 54x^{5} + 31x^{4} + 558x^{3} + 32x^{2} - 1713x - 1121 \)
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: | \(4195872914689\) \(\medspace = 127^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(63.57\) | 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: | $127^{6/7}\approx 63.57118869849136$ | ||
Ramified primes: | \(127\) | 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: | \(127\) | ||
Dirichlet character group: | $\lbrace$$\chi_{127}(32,·)$, $\chi_{127}(1,·)$, $\chi_{127}(2,·)$, $\chi_{127}(4,·)$, $\chi_{127}(16,·)$, $\chi_{127}(8,·)$, $\chi_{127}(64,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{19}a^{5}+\frac{2}{19}a^{4}+\frac{2}{19}a^{3}+\frac{4}{19}a^{2}+\frac{5}{19}a$, $\frac{1}{13357}a^{6}-\frac{275}{13357}a^{5}-\frac{4846}{13357}a^{4}+\frac{5492}{13357}a^{3}+\frac{5091}{13357}a^{2}-\frac{5774}{13357}a+\frac{223}{703}$
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)
| |
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: | $\frac{276}{13357}a^{6}-\frac{1382}{13357}a^{5}-\frac{13044}{13357}a^{4}+\frac{61988}{13357}a^{3}+\frac{86991}{13357}a^{2}-\frac{299401}{13357}a-\frac{12970}{703}$, $\frac{20}{703}a^{6}-\frac{61}{703}a^{5}-\frac{979}{703}a^{4}+\frac{2614}{703}a^{3}+\frac{362}{37}a^{2}-\frac{11658}{703}a-\frac{905}{37}$, $\frac{34}{13357}a^{6}-\frac{914}{13357}a^{5}-\frac{965}{13357}a^{4}+\frac{43316}{13357}a^{3}-\frac{6874}{13357}a^{2}-\frac{234278}{13357}a-\frac{2260}{703}$, $\frac{1359}{13357}a^{6}-\frac{3947}{13357}a^{5}-\frac{62577}{13357}a^{4}+\frac{162270}{13357}a^{3}+\frac{290082}{13357}a^{2}-\frac{615105}{13357}a-\frac{15402}{703}$, $\frac{1030}{13357}a^{6}-\frac{2753}{13357}a^{5}-\frac{49290}{13357}a^{4}+\frac{113605}{13357}a^{3}+\frac{301640}{13357}a^{2}-\frac{470850}{13357}a-\frac{25499}{703}$, $\frac{310}{13357}a^{6}-\frac{2296}{13357}a^{5}-\frac{14009}{13357}a^{4}+\frac{105304}{13357}a^{3}+\frac{80117}{13357}a^{2}-\frac{547036}{13357}a-\frac{22260}{703}$ | 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: | \( 8535.73766168 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 8535.73766168 \cdot 1}{2\cdot\sqrt{4195872914689}}\cr\approx \mathstrut & 0.266691927412 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.1.0.1}{1} }^{7}$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.1.0.1}{1} }^{7}$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.1.0.1}{1} }^{7}$ |
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 |
---|---|---|---|---|---|---|---|
\(127\) | 127.7.6.1 | $x^{7} + 127$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.127.7t1.a.a | $1$ | $ 127 $ | 7.7.4195872914689.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.127.7t1.a.b | $1$ | $ 127 $ | 7.7.4195872914689.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.127.7t1.a.c | $1$ | $ 127 $ | 7.7.4195872914689.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.127.7t1.a.d | $1$ | $ 127 $ | 7.7.4195872914689.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.127.7t1.a.e | $1$ | $ 127 $ | 7.7.4195872914689.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.127.7t1.a.f | $1$ | $ 127 $ | 7.7.4195872914689.1 | $C_7$ (as 7T1) | $0$ | $1$ |