Normalized defining polynomial
\( x^{8} - 19x^{6} + 41x^{4} - 19x^{2} + 1 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[8, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(44152515625\) \(\medspace = 5^{6}\cdot 41^{4}\) | 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: | \(21.41\) | 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: | $5^{3/4}41^{1/2}\approx 21.410136276713814$ | ||
Ramified primes: | \(5\), \(41\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{18}a^{6}-\frac{1}{9}a^{4}-\frac{1}{9}a^{2}-\frac{1}{2}a-\frac{4}{9}$, $\frac{1}{18}a^{7}-\frac{1}{9}a^{5}-\frac{1}{9}a^{3}-\frac{1}{2}a^{2}-\frac{4}{9}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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)
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Fundamental units: | $\frac{4}{9}a^{7}-\frac{151}{18}a^{5}+\frac{154}{9}a^{3}-\frac{50}{9}a-\frac{1}{2}$, $a$, $\frac{1}{18}a^{7}+\frac{5}{18}a^{6}-\frac{10}{9}a^{5}-\frac{91}{18}a^{4}+\frac{61}{18}a^{3}+\frac{67}{9}a^{2}-\frac{40}{9}a-\frac{13}{18}$, $\frac{5}{18}a^{7}-\frac{1}{9}a^{6}-\frac{91}{18}a^{5}+\frac{20}{9}a^{4}+\frac{67}{9}a^{3}-\frac{113}{18}a^{2}-\frac{11}{9}a+\frac{7}{18}$, $\frac{1}{6}a^{7}+\frac{1}{18}a^{6}-\frac{10}{3}a^{5}-\frac{10}{9}a^{4}+\frac{29}{3}a^{3}+\frac{61}{18}a^{2}-\frac{29}{6}a-\frac{13}{9}$, $\frac{1}{6}a^{7}-\frac{1}{18}a^{6}-\frac{10}{3}a^{5}+\frac{10}{9}a^{4}+\frac{29}{3}a^{3}-\frac{61}{18}a^{2}-\frac{29}{6}a+\frac{13}{9}$, $\frac{41}{9}a^{7}-\frac{5}{6}a^{6}-\frac{1559}{18}a^{5}+\frac{97}{6}a^{4}+\frac{3373}{18}a^{3}-\frac{239}{6}a^{2}-\frac{733}{9}a+\frac{59}{3}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 621.097454411 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 621.097454411 \cdot 1}{2\cdot\sqrt{44152515625}}\cr\approx \mathstrut & 0.378348479070 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $D_4$ |
Character table for $D_4$ |
Intermediate fields
\(\Q(\sqrt{41}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{205}) \), \(\Q(\sqrt{5}, \sqrt{41})\), 4.4.5125.1 x2, 4.4.210125.1 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 4 siblings: | 4.4.5125.1, 4.4.210125.1 |
Minimal sibling: | 4.4.5125.1 |
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 | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(41\) | 41.4.2.1 | $x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
41.4.2.1 | $x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.205.2t1.a.a | $1$ | $ 5 \cdot 41 $ | \(\Q(\sqrt{205}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.41.2t1.a.a | $1$ | $ 41 $ | \(\Q(\sqrt{41}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
*2 | 2.1025.4t3.a.a | $2$ | $ 5^{2} \cdot 41 $ | 8.8.44152515625.1 | $D_4$ (as 8T4) | $1$ | $2$ |