Normalized defining polynomial
\( x^{7} - x^{6} + 17x^{5} - 167x^{4} - 1045x^{3} - 3755x^{2} - 6708x - 7396 \)
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: | \(-122164218843659\) \(\medspace = -\,29^{6}\cdot 59^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(102.90\) | 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: | $29^{6/7}59^{1/2}\approx 137.6924108430833$ | ||
Ramified primes: | \(29\), \(59\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-59}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4}a^{4}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{28}a^{5}-\frac{1}{14}a^{4}-\frac{5}{14}a^{3}+\frac{13}{28}a^{2}-\frac{3}{7}a-\frac{1}{7}$, $\frac{1}{2202116}a^{6}+\frac{15931}{1101058}a^{5}-\frac{271141}{2202116}a^{4}-\frac{386135}{2202116}a^{3}+\frac{372653}{1101058}a^{2}-\frac{724091}{2202116}a+\frac{1451}{3658}$
Monogenic: | No | |
Index: | $8$ | |
Inessential primes: | $2$ |
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: | $\frac{810065}{2202116}a^{6}+\frac{940981}{2202116}a^{5}+\frac{40156937}{2202116}a^{4}+\frac{26038547}{2202116}a^{3}+\frac{354959781}{2202116}a^{2}+\frac{201840749}{2202116}a+\frac{1906941}{3658}$, $\frac{2906769}{2202116}a^{6}+\frac{4713872}{550529}a^{5}+\frac{178827185}{2202116}a^{4}+\frac{740604841}{2202116}a^{3}+\frac{511268112}{550529}a^{2}+\frac{3240312175}{2202116}a+\frac{35484455}{25606}$, $\frac{130886045}{2202116}a^{6}+\frac{393557231}{1101058}a^{5}+\frac{7739865999}{2202116}a^{4}+\frac{32432855417}{2202116}a^{3}+\frac{45280359001}{1101058}a^{2}+\frac{144264408829}{2202116}a+\frac{227811625}{3658}$ | 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: | \( 18669.7108589 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{3}\cdot 18669.7108589 \cdot 7}{2\cdot\sqrt{122164218843659}}\cr\approx \mathstrut & 2.93294031978 \end{aligned}\]
Galois group
A solvable group of order 14 |
The 5 conjugacy class representatives for $D_{7}$ |
Character table for $D_{7}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Galois closure: | deg 14 |
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 | ${\href{/padicField/2.2.0.1}{2} }^{3}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.1.0.1}{1} }^{7}$ | ${\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | R | ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | R |
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 |
---|---|---|---|---|---|---|---|
\(29\) | 29.7.6.1 | $x^{7} + 232$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
\(59\) | $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.59.2t1.a.a | $1$ | $ 59 $ | \(\Q(\sqrt{-59}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.49619.7t2.a.b | $2$ | $ 29^{2} \cdot 59 $ | 7.1.122164218843659.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
* | 2.49619.7t2.a.c | $2$ | $ 29^{2} \cdot 59 $ | 7.1.122164218843659.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
* | 2.49619.7t2.a.a | $2$ | $ 29^{2} \cdot 59 $ | 7.1.122164218843659.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |