Normalized defining polynomial
\( x^{8} - 4x^{7} + 68x^{6} - 190x^{5} + 1259x^{4} - 2206x^{3} - 832x^{2} + 1904x + 1744 \)
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: | \(816189189450625\) \(\medspace = 5^{4}\cdot 1069^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(73.11\) | 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: | $5^{1/2}1069^{1/2}\approx 73.10950690573696$ | ||
Ramified primes: | \(5\), \(1069\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | 4.0.26725.1$^{4}$, 4.0.5713805.1$^{4}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{3}$, $\frac{1}{3560}a^{6}-\frac{3}{3560}a^{5}-\frac{309}{3560}a^{4}+\frac{7}{40}a^{3}-\frac{4}{445}a^{2}-\frac{7}{89}a+\frac{22}{445}$, $\frac{1}{544680}a^{7}+\frac{73}{544680}a^{6}+\frac{25273}{544680}a^{5}-\frac{19301}{544680}a^{4}+\frac{44573}{272340}a^{3}-\frac{233}{136170}a^{2}-\frac{38651}{136170}a-\frac{28588}{68085}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
$C_{3}\times C_{3}$, which has order $9$
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: | $\frac{91}{272340}a^{7}-\frac{637}{544680}a^{6}+\frac{2335}{108936}a^{5}-\frac{5519}{108936}a^{4}+\frac{197363}{544680}a^{3}-\frac{33596}{68085}a^{2}-\frac{136477}{136170}a+\frac{5563}{68085}$, $\frac{3643339}{54468}a^{7}+\frac{65923483}{544680}a^{6}+\frac{1023639691}{544680}a^{5}+\frac{427603813}{544680}a^{4}-\frac{6864412901}{544680}a^{3}+\frac{403837}{765}a^{2}+\frac{362281829}{27234}a+\frac{576302321}{68085}$, $\frac{3643339}{54468}a^{7}-\frac{320957213}{544680}a^{6}+\frac{2184281779}{544680}a^{5}-\frac{7809823163}{544680}a^{4}+\frac{7676037571}{544680}a^{3}+\frac{718716692}{68085}a^{2}-\frac{264667495}{27234}a-\frac{854096821}{68085}$ | 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: | \( 2078.16075839 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 2078.16075839 \cdot 9}{2\cdot\sqrt{816189189450625}}\cr\approx \mathstrut & 0.510170929383 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $D_4$ |
Character table for $D_4$ |
Intermediate fields
\(\Q(\sqrt{5345}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{1069}) \), \(\Q(\sqrt{5}, \sqrt{1069})\), 4.0.26725.1 x2, 4.0.5713805.1 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 4 siblings: | 4.0.5713805.1, 4.0.26725.1 |
Minimal sibling: | 4.0.26725.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.1.0.1}{1} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(1069\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.1069.2t1.a.a | $1$ | $ 1069 $ | \(\Q(\sqrt{1069}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.5345.2t1.a.a | $1$ | $ 5 \cdot 1069 $ | \(\Q(\sqrt{5345}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
*2 | 2.5345.4t3.c.a | $2$ | $ 5 \cdot 1069 $ | 8.0.816189189450625.1 | $D_4$ (as 8T4) | $1$ | $-2$ |