Normalized defining polynomial
\( x^{7} - 7x^{5} - 14x^{4} - 7x^{3} - 7x + 2 \)
Invariants
Degree: | $7$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1475789056\) \(\medspace = 2^{8}\cdot 7^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(20.41\) | 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: | $2^{2}7^{26/21}\approx 44.50121314235103$ | ||
Ramified primes: | \(2\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{32}a^{6}+\frac{5}{32}a^{5}-\frac{7}{16}a^{4}+\frac{3}{8}a^{3}-\frac{11}{32}a^{2}+\frac{9}{32}a+\frac{3}{16}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | 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: | $\frac{1}{32}a^{6}+\frac{5}{32}a^{5}-\frac{7}{16}a^{4}-\frac{13}{8}a^{3}-\frac{11}{32}a^{2}+\frac{41}{32}a+\frac{3}{16}$, $\frac{23}{32}a^{6}+\frac{3}{32}a^{5}-\frac{81}{16}a^{4}-\frac{83}{8}a^{3}-\frac{221}{32}a^{2}-\frac{65}{32}a-\frac{75}{16}$, $\frac{5}{16}a^{6}-\frac{7}{16}a^{5}-\frac{11}{8}a^{4}-\frac{9}{4}a^{3}-\frac{23}{16}a^{2}-\frac{3}{16}a-\frac{9}{8}$, $\frac{103}{32}a^{6}+\frac{51}{32}a^{5}-\frac{353}{16}a^{4}-\frac{451}{8}a^{3}-\frac{1549}{32}a^{2}-\frac{529}{32}a-\frac{347}{16}$ | 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: | \( 326.439203548 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{2}\cdot 326.439203548 \cdot 1}{2\cdot\sqrt{1475789056}}\cr\approx \mathstrut & 1.34186830488 \end{aligned}\]
Galois group
$\GL(3,2)$ (as 7T5):
A non-solvable group of order 168 |
The 6 conjugacy class representatives for $\GL(3,2)$ |
Character table for $\GL(3,2)$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 8 sibling: | 8.0.377801998336.3 |
Degree 14 siblings: | deg 14, deg 14 |
Degree 21 sibling: | deg 21 |
Degree 24 sibling: | deg 24 |
Degree 28 sibling: | deg 28 |
Degree 42 siblings: | deg 42, deg 42, deg 42 |
Arithmetically equvalently sibling: | 7.3.1475789056.1 |
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 | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | R | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
2.4.8.6 | $x^{4} + 6 x^{2} + 4 x + 10$ | $4$ | $1$ | $8$ | $D_{4}$ | $[2, 3]^{2}$ | |
\(7\) | 7.7.8.1 | $x^{7} + 14 x^{2} + 7$ | $7$ | $1$ | $8$ | $C_7:C_3$ | $[4/3]_{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
3.614656.42t37.a.a | $3$ | $ 2^{8} \cdot 7^{4}$ | 7.3.1475789056.2 | $\GL(3,2)$ (as 7T5) | $0$ | $-1$ | |
3.614656.42t37.a.b | $3$ | $ 2^{8} \cdot 7^{4}$ | 7.3.1475789056.2 | $\GL(3,2)$ (as 7T5) | $0$ | $-1$ | |
* | 6.1475789056.7t5.a.a | $6$ | $ 2^{8} \cdot 7^{8}$ | 7.3.1475789056.2 | $\GL(3,2)$ (as 7T5) | $1$ | $2$ |
7.377801998336.8t37.a.a | $7$ | $ 2^{16} \cdot 7^{8}$ | 7.3.1475789056.2 | $\GL(3,2)$ (as 7T5) | $1$ | $-1$ | |
8.185...464.21t14.a.a | $8$ | $ 2^{16} \cdot 7^{10}$ | 7.3.1475789056.2 | $\GL(3,2)$ (as 7T5) | $1$ | $0$ |