Normalized defining polynomial
\( x^{8} - 4x^{7} + 4x^{6} + 2x^{5} - 40x^{4} + 72x^{3} + x^{2} - 36x + 51 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(5554571841\) \(\medspace = 3^{4}\cdot 7^{4}\cdot 13^{4}\) | 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: | \(16.52\) | 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: | $3^{1/2}7^{1/2}13^{1/2}\approx 16.522711641858304$ | ||
Ramified primes: | \(3\), \(7\), \(13\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{62}a^{6}-\frac{3}{62}a^{5}-\frac{7}{31}a^{4}-\frac{29}{62}a^{3}-\frac{7}{31}a^{2}-\frac{3}{62}a+\frac{11}{31}$, $\frac{1}{6262}a^{7}+\frac{47}{6262}a^{6}-\frac{629}{6262}a^{5}-\frac{535}{3131}a^{4}+\frac{167}{3131}a^{3}-\frac{739}{3131}a^{2}+\frac{742}{3131}a+\frac{201}{6262}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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)
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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: | $\frac{23}{3131}a^{7}-\frac{131}{3131}a^{6}+\frac{255}{6262}a^{5}+\frac{371}{6262}a^{4}-\frac{1004}{3131}a^{3}+\frac{6651}{6262}a^{2}+\frac{196}{3131}a-\frac{109}{202}$, $\frac{65}{3131}a^{7}-\frac{455}{6262}a^{6}+\frac{545}{6262}a^{5}-\frac{225}{6262}a^{4}-\frac{4151}{6262}a^{3}+\frac{3112}{3131}a^{2}-\frac{1712}{3131}a+\frac{3809}{6262}$, $\frac{2}{31}a^{6}-\frac{6}{31}a^{5}+\frac{3}{31}a^{4}+\frac{4}{31}a^{3}-\frac{59}{31}a^{2}+\frac{56}{31}a+\frac{106}{31}$, $\frac{66}{3131}a^{7}-\frac{231}{3131}a^{6}-\frac{205}{3131}a^{5}+\frac{1090}{3131}a^{4}-\frac{277}{3131}a^{3}-\frac{790}{3131}a^{2}+\frac{1489}{3131}a-\frac{571}{3131}$, $\frac{229}{6262}a^{7}-\frac{426}{3131}a^{6}+\frac{389}{6262}a^{5}+\frac{2117}{6262}a^{4}-\frac{12495}{6262}a^{3}+\frac{21401}{6262}a^{2}-\frac{1039}{6262}a-\frac{2993}{3131}$ | 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: | \( 127.69841012 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 127.69841012 \cdot 1}{2\cdot\sqrt{5554571841}}\cr\approx \mathstrut & 0.54114035204 \end{aligned}\]
Galois group
$C_2\times D_4$ (as 8T9):
A solvable group of order 16 |
The 10 conjugacy class representatives for $D_4\times C_2$ |
Character table for $D_4\times C_2$ |
Intermediate fields
\(\Q(\sqrt{13}) \), \(\Q(\sqrt{273}) \), \(\Q(\sqrt{21}) \), 4.2.507.1, 4.2.24843.1, \(\Q(\sqrt{13}, \sqrt{21})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 16.0.30853268336830129281.1 |
Degree 8 siblings: | 8.0.617174649.1, 8.0.32867289.1, 8.0.5554571841.3 |
Minimal sibling: | 8.0.32867289.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(7\) | 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(13\) | 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |