Normalized defining polynomial
\( x^{8} + 7x^{6} - 11x^{5} + 65x^{4} + 43x^{3} + 191x^{2} + 137x + 240 \)
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: | \(2528484794641\) \(\medspace = 13^{4}\cdot 97^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(35.51\) | 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: | $13^{2/3}97^{2/3}\approx 116.71949263039971$ | ||
Ramified primes: | \(13\), \(97\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5804911}a^{7}+\frac{2450864}{5804911}a^{6}-\frac{1799145}{5804911}a^{5}-\frac{2876403}{5804911}a^{4}-\frac{2886575}{5804911}a^{3}-\frac{982460}{5804911}a^{2}+\frac{176793}{829273}a-\frac{2608210}{5804911}$
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: | $\frac{869051}{5804911}a^{7}-\frac{524234}{5804911}a^{6}+\frac{4016455}{5804911}a^{5}-\frac{12714000}{5804911}a^{4}+\frac{52032702}{5804911}a^{3}+\frac{30708619}{5804911}a^{2}+\frac{3554006}{829273}a-\frac{87764561}{5804911}$, $\frac{417458}{5804911}a^{7}-\frac{194771}{5804911}a^{6}+\frac{936325}{5804911}a^{5}-\frac{8383580}{5804911}a^{4}+\frac{17648140}{5804911}a^{3}+\frac{27614758}{5804911}a^{2}+\frac{5818651}{829273}a-\frac{8388643}{5804911}$, $\frac{182878}{5804911}a^{7}+\frac{318460}{5804911}a^{6}-\frac{1683830}{5804911}a^{5}+\frac{4402075}{5804911}a^{4}+\frac{1738579}{5804911}a^{3}+\frac{14895214}{5804911}a^{2}+\frac{1513076}{829273}a+\frac{22723223}{5804911}$ | 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: | \( 1419.45248175 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 1419.45248175 \cdot 2}{2\cdot\sqrt{2528484794641}}\cr\approx \mathstrut & 1.39126595788 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.6.4.2 | $x^{6} - 156 x^{3} + 338$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
\(97\) | $\Q_{97}$ | $x + 92$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{97}$ | $x + 92$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
97.3.2.2 | $x^{3} + 194$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
97.3.2.2 | $x^{3} + 194$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.1261.3t1.a.a | $1$ | $ 13 \cdot 97 $ | 3.3.1590121.2 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.1261.3t1.a.b | $1$ | $ 13 \cdot 97 $ | 3.3.1590121.2 | $C_3$ (as 3T1) | $0$ | $1$ | |
3.1590121.7t3.a.a | $3$ | $ 13^{2} \cdot 97^{2}$ | 7.7.2528484794641.1 | $C_7:C_3$ (as 7T3) | $0$ | $3$ | |
3.1590121.7t3.a.b | $3$ | $ 13^{2} \cdot 97^{2}$ | 7.7.2528484794641.1 | $C_7:C_3$ (as 7T3) | $0$ | $3$ | |
* | 7.252...641.8t36.d.a | $7$ | $ 13^{4} \cdot 97^{4}$ | 8.0.2528484794641.4 | $C_2^3:(C_7: C_3)$ (as 8T36) | $1$ | $-1$ |
7.318...301.24t283.d.a | $7$ | $ 13^{5} \cdot 97^{5}$ | 8.0.2528484794641.4 | $C_2^3:(C_7: C_3)$ (as 8T36) | $0$ | $-1$ | |
7.318...301.24t283.d.b | $7$ | $ 13^{5} \cdot 97^{5}$ | 8.0.2528484794641.4 | $C_2^3:(C_7: C_3)$ (as 8T36) | $0$ | $-1$ |