Normalized defining polynomial
\( x^{8} - 2x^{7} + 2x^{6} - 8x^{5} + 60x^{4} - 60x^{3} - 52x^{2} + 166x + 235 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(4355602392064\) \(\medspace = 2^{10}\cdot 7^{4}\cdot 11^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(38.01\) | 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: | $2^{19/14}7^{2/3}11^{6/7}\approx 73.20853283844976$ | ||
Ramified primes: | \(2\), \(7\), \(11\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{766439}a^{7}+\frac{86360}{766439}a^{6}+\frac{4413}{766439}a^{5}+\frac{195315}{766439}a^{4}+\frac{4578}{766439}a^{3}-\frac{117348}{766439}a^{2}+\frac{214869}{766439}a+\frac{262115}{766439}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
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: | $\frac{38081}{766439}a^{7}-\frac{114589}{766439}a^{6}+\frac{201312}{766439}a^{5}-\frac{499980}{766439}a^{4}+\frac{2652482}{766439}a^{3}-\frac{4222013}{766439}a^{2}+\frac{690064}{766439}a+\frac{7930608}{766439}$, $\frac{101910}{766439}a^{7}-\frac{71437}{766439}a^{6}-\frac{170863}{766439}a^{5}-\frac{635619}{766439}a^{4}+\frac{2848385}{766439}a^{3}-\frac{953402}{766439}a^{2}-\frac{9059708}{766439}a-\frac{8223207}{766439}$, $\frac{518178}{766439}a^{7}-\frac{221813}{766439}a^{6}+\frac{431977}{766439}a^{5}-\frac{6465392}{766439}a^{4}+\frac{13886081}{766439}a^{3}+\frac{20512852}{766439}a^{2}-\frac{6336360}{766439}a-\frac{22954768}{766439}$ | 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: | \( 2770.51838232 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 2770.51838232 \cdot 2}{2\cdot\sqrt{4355602392064}}\cr\approx \mathstrut & 2.06898049123 \end{aligned}\]
Galois group
A solvable group of order 168 |
The 8 conjugacy class representatives for $C_2^3:(C_7: C_3)$ |
Character table for $C_2^3:(C_7: C_3)$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 14 sibling: | deg 14 |
Degree 24 sibling: | deg 24 |
Degree 28 sibling: | deg 28 |
Degree 42 sibling: | 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 | R | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | R | ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.10.1 | $x^{8} + 2 x^{3} + 2$ | $8$ | $1$ | $10$ | $C_2^3:(C_7: C_3)$ | $[10/7, 10/7, 10/7]_{7}^{3}$ |
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
\(11\) | $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
11.7.6.1 | $x^{7} + 11$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
3.521752.7t3.a.a | $3$ | $ 2^{3} \cdot 7^{2} \cdot 11^{3}$ | 7.7.272225149504.1 | $C_7:C_3$ (as 7T3) | $0$ | $3$ | |
3.521752.7t3.a.b | $3$ | $ 2^{3} \cdot 7^{2} \cdot 11^{3}$ | 7.7.272225149504.1 | $C_7:C_3$ (as 7T3) | $0$ | $3$ | |
* | 7.435...064.8t36.b.a | $7$ | $ 2^{10} \cdot 7^{4} \cdot 11^{6}$ | 8.0.4355602392064.4 | $C_2^3:(C_7: C_3)$ (as 8T36) | $1$ | $-1$ |
7.304...448.24t283.b.a | $7$ | $ 2^{10} \cdot 7^{5} \cdot 11^{6}$ | 8.0.4355602392064.4 | $C_2^3:(C_7: C_3)$ (as 8T36) | $0$ | $-1$ | |
7.304...448.24t283.b.b | $7$ | $ 2^{10} \cdot 7^{5} \cdot 11^{6}$ | 8.0.4355602392064.4 | $C_2^3:(C_7: C_3)$ (as 8T36) | $0$ | $-1$ |