Normalized defining polynomial
\( x^{8} - 35x^{6} + 257x^{4} - 140x^{2} + 16 \)
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: | \(1718786550625\) \(\medspace = 5^{4}\cdot 229^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(33.84\) | 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: | $5^{1/2}229^{1/2}\approx 33.83784863137726$ | ||
Ramified primes: | \(5\), \(229\) | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{360}a^{6}-\frac{1}{4}a^{5}+\frac{17}{360}a^{4}-\frac{1}{4}a^{3}-\frac{119}{360}a^{2}+\frac{1}{4}a-\frac{7}{90}$, $\frac{1}{720}a^{7}+\frac{17}{720}a^{5}+\frac{61}{720}a^{3}+\frac{19}{90}a-\frac{1}{2}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
$C_{3}$, which has order $3$
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)
| |
Fundamental units: | $\frac{7}{180}a^{6}-\frac{241}{180}a^{4}+\frac{1687}{180}a^{2}-\frac{94}{45}$, $\frac{31}{720}a^{7}+\frac{7}{120}a^{6}-\frac{1093}{720}a^{5}-\frac{241}{120}a^{4}+\frac{8191}{720}a^{3}+\frac{1687}{120}a^{2}-\frac{1297}{180}a-\frac{62}{15}$, $\frac{31}{720}a^{7}+\frac{7}{360}a^{6}-\frac{1093}{720}a^{5}-\frac{241}{360}a^{4}+\frac{8191}{720}a^{3}+\frac{1687}{360}a^{2}-\frac{1297}{180}a-\frac{137}{45}$, $\frac{47}{360}a^{7}-\frac{7}{120}a^{6}-\frac{1631}{360}a^{5}+\frac{241}{120}a^{4}+\frac{11597}{360}a^{3}-\frac{1687}{120}a^{2}-\frac{1693}{180}a+\frac{109}{30}$, $\frac{7}{144}a^{7}-\frac{7}{90}a^{6}-\frac{241}{144}a^{5}+\frac{241}{90}a^{4}+\frac{1651}{144}a^{3}-\frac{1687}{90}a^{2}+\frac{8}{9}a+\frac{331}{90}$, $\frac{25}{144}a^{7}-\frac{871}{144}a^{5}+\frac{6277}{144}a^{3}-\frac{299}{18}a+\frac{1}{2}$, $\frac{1}{90}a^{6}-\frac{14}{45}a^{4}+\frac{61}{90}a^{2}+\frac{1}{2}a-\frac{14}{45}$ | 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: | \( 1659.15027948 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 1659.15027948 \cdot 3}{2\cdot\sqrt{1718786550625}}\cr\approx \mathstrut & 0.485966100814 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $D_4$ |
Character table for $D_4$ |
Intermediate fields
\(\Q(\sqrt{1145}) \), \(\Q(\sqrt{229}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{229})\), 4.4.262205.1 x2, 4.4.5725.1 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 4 siblings: | 4.4.262205.1, 4.4.5725.1 |
Minimal sibling: | 4.4.5725.1 |
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 | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.1.0.1}{1} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(229\) | Deg $4$ | $2$ | $2$ | $2$ | |||
Deg $4$ | $2$ | $2$ | $2$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.229.2t1.a.a | $1$ | $ 229 $ | \(\Q(\sqrt{229}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.1145.2t1.a.a | $1$ | $ 5 \cdot 229 $ | \(\Q(\sqrt{1145}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
*2 | 2.1145.4t3.c.a | $2$ | $ 5 \cdot 229 $ | 8.8.1718786550625.1 | $D_4$ (as 8T4) | $1$ | $2$ |