Normalized defining polynomial
\( x^{7} - x^{6} - 7x^{5} + 2x^{4} + 12x^{3} - 5x - 1 \)
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)
| |
Discriminant: | \(55078981\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.76\) | 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: | $55078981^{1/2}\approx 7421.521474738182$ | ||
Ramified primes: | \(55078981\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{55078981}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$
Monogenic: | Yes | |
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: | $a$, $a^{5}-2a^{4}-4a^{3}+5a^{2}+2a-1$, $a^{6}-2a^{5}-5a^{4}+7a^{3}+5a^{2}-5a$, $a^{5}-a^{4}-6a^{3}+a^{2}+7a+1$, $a^{5}-a^{4}-6a^{3}+a^{2}+6a+1$, $a^{6}-a^{5}-7a^{4}+3a^{3}+10a^{2}-4a-1$ | 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: | \( 25.8498011328 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 25.8498011328 \cdot 1}{2\cdot\sqrt{55078981}}\cr\approx \mathstrut & 0.222917534919 \end{aligned}\]
Galois group
A non-solvable group of order 5040 |
The 15 conjugacy class representatives for $S_7$ |
Character table for $S_7$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 14 sibling: | deg 14 |
Degree 21 sibling: | deg 21 |
Degree 30 sibling: | deg 30 |
Degree 35 sibling: | deg 35 |
Degree 42 siblings: | deg 42, deg 42, deg 42, deg 42 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(55078981\) | $\Q_{55078981}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.55078981.2t1.a.a | $1$ | $ 55078981 $ | \(\Q(\sqrt{55078981}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
6.506...901.14t46.a.a | $6$ | $ 55078981^{5}$ | 7.7.55078981.1 | $S_7$ (as 7T7) | $1$ | $6$ | |
* | 6.55078981.7t7.a.a | $6$ | $ 55078981 $ | 7.7.55078981.1 | $S_7$ (as 7T7) | $1$ | $6$ |
14.920...321.21t38.a.a | $14$ | $ 55078981^{4}$ | 7.7.55078981.1 | $S_7$ (as 7T7) | $1$ | $14$ | |
14.256...801.42t413.a.a | $14$ | $ 55078981^{10}$ | 7.7.55078981.1 | $S_7$ (as 7T7) | $1$ | $14$ | |
14.466...221.30t565.a.a | $14$ | $ 55078981^{9}$ | 7.7.55078981.1 | $S_7$ (as 7T7) | $1$ | $14$ | |
14.506...901.30t565.a.a | $14$ | $ 55078981^{5}$ | 7.7.55078981.1 | $S_7$ (as 7T7) | $1$ | $14$ | |
15.506...901.42t412.a.a | $15$ | $ 55078981^{5}$ | 7.7.55078981.1 | $S_7$ (as 7T7) | $1$ | $15$ | |
15.256...801.42t411.a.a | $15$ | $ 55078981^{10}$ | 7.7.55078981.1 | $S_7$ (as 7T7) | $1$ | $15$ | |
20.256...801.70.a.a | $20$ | $ 55078981^{10}$ | 7.7.55078981.1 | $S_7$ (as 7T7) | $1$ | $20$ | |
21.256...801.84.a.a | $21$ | $ 55078981^{10}$ | 7.7.55078981.1 | $S_7$ (as 7T7) | $1$ | $21$ | |
21.141...781.42t418.a.a | $21$ | $ 55078981^{11}$ | 7.7.55078981.1 | $S_7$ (as 7T7) | $1$ | $21$ | |
35.660...601.126.a.a | $35$ | $ 55078981^{20}$ | 7.7.55078981.1 | $S_7$ (as 7T7) | $1$ | $35$ | |
35.130...701.70.a.a | $35$ | $ 55078981^{15}$ | 7.7.55078981.1 | $S_7$ (as 7T7) | $1$ | $35$ |