Normalized defining polynomial
\( x^{8} - 4x^{7} + 8x^{6} - 2x^{5} + 29x^{4} - 62x^{3} + 18x^{2} - 84x + 196 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(4569760000\) \(\medspace = 2^{8}\cdot 5^{4}\cdot 13^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(16.12\) | 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))
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Galois root discriminant: | $2\cdot 5^{1/2}13^{1/2}\approx 16.1245154965971$ | ||
Ramified primes: | \(2\), \(5\), \(13\) | 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 not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{14}a^{5}-\frac{3}{14}a^{4}+\frac{1}{14}a^{3}-\frac{3}{14}a^{2}-\frac{3}{7}a$, $\frac{1}{70}a^{6}-\frac{1}{35}a^{5}-\frac{1}{35}a^{4}+\frac{13}{35}a^{3}+\frac{33}{70}a^{2}+\frac{4}{35}a-\frac{1}{5}$, $\frac{1}{6790}a^{7}+\frac{43}{6790}a^{6}-\frac{101}{3395}a^{5}+\frac{43}{970}a^{4}+\frac{2633}{6790}a^{3}+\frac{1559}{3395}a^{2}+\frac{1308}{3395}a-\frac{24}{97}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{2}$, which has order $4$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{2}{3395} a^{7} - \frac{15}{1358} a^{6} + \frac{129}{6790} a^{5} + \frac{57}{6790} a^{4} - \frac{341}{1358} a^{3} - \frac{513}{3395} a^{2} + \frac{6}{3395} a + \frac{383}{485} \) (order $4$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $\frac{36}{3395}a^{7}-\frac{299}{6790}a^{6}+\frac{491}{6790}a^{5}+\frac{166}{3395}a^{4}+\frac{911}{6790}a^{3}-\frac{3939}{6790}a^{2}+\frac{1541}{3395}a+\frac{18}{97}$, $\frac{44}{3395}a^{7}-\frac{193}{6790}a^{6}+\frac{363}{6790}a^{5}+\frac{246}{3395}a^{4}+\frac{2687}{6790}a^{3}+\frac{79}{970}a^{2}+\frac{741}{3395}a-\frac{278}{485}$, $\frac{19}{1358}a^{7}+\frac{11}{6790}a^{6}-\frac{186}{3395}a^{5}+\frac{669}{3395}a^{4}+\frac{5501}{6790}a^{3}+\frac{4143}{6790}a^{2}-\frac{5526}{3395}a-\frac{1021}{485}$ | 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: | \( 38.1030791243 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 38.1030791243 \cdot 4}{4\cdot\sqrt{4569760000}}\cr\approx \mathstrut & 0.878481965230 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $D_4$ |
Character table for $D_4$ |
Intermediate fields
\(\Q(\sqrt{-65}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{65})\), 4.2.16900.1 x2, 4.0.1040.1 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 4 siblings: | 4.2.16900.1, 4.0.1040.1 |
Minimal sibling: | 4.0.1040.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
\(5\) | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(13\) | 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.65.2t1.a.a | $1$ | $ 5 \cdot 13 $ | \(\Q(\sqrt{65}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.260.2t1.a.a | $1$ | $ 2^{2} \cdot 5 \cdot 13 $ | \(\Q(\sqrt{-65}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
*2 | 2.260.4t3.d.a | $2$ | $ 2^{2} \cdot 5 \cdot 13 $ | 8.0.4569760000.4 | $D_4$ (as 8T4) | $1$ | $0$ |