Normalized defining polynomial
\( x^{7} - 882x^{4} + 882x^{3} - 15876 \)
Invariants
Degree: | $7$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-247979172729544704\) \(\medspace = -\,2^{13}\cdot 3^{7}\cdot 7^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(305.43\) | 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: | $2^{37/12}3^{11/6}7^{12/7}\approx 1784.9975216126422$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-6}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{126}a^{4}$, $\frac{1}{630}a^{5}+\frac{1}{315}a^{4}-\frac{1}{15}a^{3}+\frac{1}{15}a^{2}-\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{9450}a^{6}+\frac{1}{1575}a^{5}+\frac{2}{525}a^{4}+\frac{3}{25}a^{3}+\frac{11}{75}a^{2}-\frac{3}{25}a+\frac{7}{25}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{7}$, which has order $7$
Unit group
Rank: | $3$ | 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: | $\frac{63\!\cdots\!89}{4725}a^{6}+\frac{65\!\cdots\!76}{525}a^{5}+\frac{18\!\cdots\!93}{1575}a^{4}-\frac{31\!\cdots\!91}{25}a^{3}+\frac{675807516569111}{25}a^{2}+\frac{62\!\cdots\!91}{25}a+\frac{57\!\cdots\!21}{25}$, $\frac{17\!\cdots\!67}{9450}a^{6}-\frac{17\!\cdots\!87}{1050}a^{5}-\frac{41\!\cdots\!18}{1575}a^{4}-\frac{12\!\cdots\!52}{75}a^{3}+\frac{76\!\cdots\!89}{25}a^{2}+\frac{21\!\cdots\!14}{25}a-\frac{32\!\cdots\!11}{25}$, $\frac{33\!\cdots\!07}{9450}a^{6}+\frac{33\!\cdots\!13}{1050}a^{5}+\frac{15\!\cdots\!89}{525}a^{4}-\frac{24\!\cdots\!62}{75}a^{3}+\frac{54\!\cdots\!77}{75}a^{2}+\frac{16\!\cdots\!29}{25}a+\frac{15\!\cdots\!49}{25}$ | 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: | \( 1346046.93683 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{3}\cdot 1346046.93683 \cdot 7}{2\cdot\sqrt{247979172729544704}}\cr\approx \mathstrut & 4.69342892735 \end{aligned}\]
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, some data not computed |
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 | R | R | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
In the table, R denotes a ramified prime. 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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.4.11.6 | $x^{4} + 18$ | $4$ | $1$ | $11$ | $D_{4}$ | $[2, 3, 4]$ | |
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.3.5.3 | $x^{3} + 9 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ | |
\(7\) | 7.7.12.1 | $x^{7} + 42 x^{6} + 7$ | $7$ | $1$ | $12$ | $C_7$ | $[2]$ |