Normalized defining polynomial
\( x^{6} - x^{5} - 65x^{4} - 160x^{3} + 20x^{2} + 208x + 64 \)
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[6, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(95388992557\) \(\medspace = 157^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(67.60\) | 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: | $157^{5/6}\approx 67.5952808785752$ | ||
Ramified primes: | \(157\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{157}) \) | ||
$\card{ \Gal(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(157\) | ||
Dirichlet character group: | $\lbrace$$\chi_{157}(144,·)$, $\chi_{157}(1,·)$, $\chi_{157}(145,·)$, $\chi_{157}(156,·)$, $\chi_{157}(12,·)$, $\chi_{157}(13,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{24}a^{4}-\frac{1}{8}a^{3}-\frac{1}{8}a^{2}-\frac{5}{12}a-\frac{1}{3}$, $\frac{1}{144}a^{5}+\frac{1}{144}a^{4}+\frac{1}{16}a^{3}-\frac{35}{72}a^{2}-\frac{1}{3}a-\frac{2}{9}$
Monogenic: | No | |
Index: | $4$ | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $\frac{1}{144}a^{5}-\frac{5}{144}a^{4}-\frac{5}{16}a^{3}+\frac{5}{36}a^{2}+\frac{7}{12}a+\frac{1}{9}$, $\frac{5}{72}a^{5}-\frac{1}{18}a^{4}-\frac{9}{2}a^{3}-\frac{863}{72}a^{2}-\frac{31}{12}a+\frac{70}{9}$, $\frac{5}{72}a^{5}-\frac{2}{9}a^{4}-4a^{3}-\frac{179}{72}a^{2}+\frac{61}{12}a+\frac{28}{9}$, $\frac{137}{144}a^{5}-\frac{187}{144}a^{4}-\frac{979}{16}a^{3}-\frac{9457}{72}a^{2}+\frac{193}{3}a+\frac{1589}{9}$, $\frac{77}{144}a^{5}-\frac{217}{144}a^{4}-\frac{497}{16}a^{3}-\frac{1127}{36}a^{2}+\frac{43}{4}a+\frac{56}{9}$ | 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: | \( 3126.56752592 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 3126.56752592 \cdot 1}{2\cdot\sqrt{95388992557}}\cr\approx \mathstrut & 0.323943044938 \end{aligned}\]
Galois group
A cyclic group of order 6 |
The 6 conjugacy class representatives for $C_6$ |
Character table for $C_6$ |
Intermediate fields
\(\Q(\sqrt{157}) \), 3.3.24649.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | 3.3.24649.1 $\times$ \(\Q(\sqrt{157}) \) $\times$ \(\Q\) |
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 | ${\href{/padicField/2.2.0.1}{2} }^{3}$ | ${\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(157\) | 157.6.5.1 | $x^{6} + 157$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.157.2t1.a.a | $1$ | $ 157 $ | \(\Q(\sqrt{157}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.157.3t1.a.a | $1$ | $ 157 $ | 3.3.24649.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.157.6t1.a.a | $1$ | $ 157 $ | 6.6.95388992557.1 | $C_6$ (as 6T1) | $0$ | $1$ |
* | 1.157.3t1.a.b | $1$ | $ 157 $ | 3.3.24649.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.157.6t1.a.b | $1$ | $ 157 $ | 6.6.95388992557.1 | $C_6$ (as 6T1) | $0$ | $1$ |