Normalized defining polynomial
\( x^{7} - 3x^{6} + 22x^{5} - 1576x^{4} + 15353x^{3} - 79092x^{2} + 197245x - 270275 \)
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)
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Discriminant: | \(-51051185753021063\) \(\medspace = -\,23^{3}\cdot 127^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(243.70\) | 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: | $23^{1/2}127^{6/7}\approx 304.8767107346861$ | ||
Ramified primes: | \(23\), \(127\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{2}+\frac{2}{5}a$, $\frac{1}{79434304145}a^{6}+\frac{1014586932}{11347757735}a^{5}-\frac{37018589292}{79434304145}a^{4}+\frac{32070471064}{79434304145}a^{3}-\frac{38406168394}{79434304145}a^{2}+\frac{26641747052}{79434304145}a+\frac{3166276720}{15886860829}$
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: | $\frac{10797037232454}{79434304145}a^{6}-\frac{9449798958982}{11347757735}a^{5}-\frac{90211416358178}{79434304145}a^{4}-\frac{18\!\cdots\!79}{79434304145}a^{3}+\frac{21\!\cdots\!04}{79434304145}a^{2}-\frac{70\!\cdots\!52}{79434304145}a+\frac{26\!\cdots\!37}{15886860829}$, $\frac{24697216851771}{15886860829}a^{6}-\frac{9182731534742}{2269551547}a^{5}+\frac{326485679280791}{15886860829}a^{4}-\frac{38\!\cdots\!30}{15886860829}a^{3}+\frac{36\!\cdots\!75}{15886860829}a^{2}-\frac{15\!\cdots\!91}{15886860829}a+\frac{24\!\cdots\!52}{15886860829}$, $\frac{5360077576856}{79434304145}a^{6}-\frac{8597129334361}{11347757735}a^{5}+\frac{7604234178197}{15886860829}a^{4}-\frac{91\!\cdots\!36}{79434304145}a^{3}+\frac{14\!\cdots\!17}{79434304145}a^{2}-\frac{15\!\cdots\!12}{15886860829}a+\frac{38\!\cdots\!37}{15886860829}$ | 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: | \( 46400.23606080308 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{3}\cdot 46400.23606080308 \cdot 7}{2\cdot\sqrt{51051185753021063}}\cr\approx \mathstrut & 0.356578306644139 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.2.0.1}{2} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.2.0.1}{2} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | R | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(23\) | $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(127\) | 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 | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.23.2t1.a.a | $1$ | $ 23 $ | \(\Q(\sqrt{-23}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.370967.7t2.a.c | $2$ | $ 23 \cdot 127^{2}$ | 7.1.51051185753021063.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
* | 2.370967.7t2.a.a | $2$ | $ 23 \cdot 127^{2}$ | 7.1.51051185753021063.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
* | 2.370967.7t2.a.b | $2$ | $ 23 \cdot 127^{2}$ | 7.1.51051185753021063.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |