Normalized defining polynomial
\( x^{10} - 10x^{8} + 40x^{6} - 100x^{4} - 80x^{3} + 100x^{2} - 320 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(576000000000000\) \(\medspace = 2^{18}\cdot 3^{2}\cdot 5^{12}\) | 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: | \(29.93\) | 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: | $2^{51/16}3^{1/2}5^{271/200}\approx 139.7005430784032$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{4}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{12}a^{8}-\frac{1}{12}a^{7}-\frac{1}{12}a^{6}+\frac{1}{6}a^{5}-\frac{1}{6}a^{3}-\frac{1}{6}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{1240992}a^{9}+\frac{1057}{310248}a^{8}+\frac{14737}{206832}a^{7}+\frac{4283}{51708}a^{6}+\frac{6323}{155124}a^{5}+\frac{13231}{77562}a^{4}+\frac{24613}{103416}a^{3}-\frac{2704}{38781}a^{2}-\frac{40327}{310248}a-\frac{18247}{77562}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | $\frac{35}{8928}a^{9}-\frac{19}{2232}a^{8}-\frac{15}{496}a^{7}+\frac{5}{93}a^{6}+\frac{151}{1116}a^{5}-\frac{55}{558}a^{4}-\frac{115}{248}a^{3}-\frac{25}{558}a^{2}+\frac{2155}{2232}a-\frac{479}{558}$, $\frac{9733}{1240992}a^{9}-\frac{2111}{310248}a^{8}-\frac{6669}{68944}a^{7}+\frac{2741}{25854}a^{6}+\frac{60947}{155124}a^{5}-\frac{53159}{77562}a^{4}-\frac{24593}{34472}a^{3}+\frac{93169}{77562}a^{2}-\frac{142387}{310248}a-\frac{110341}{77562}$, $\frac{17837}{137888}a^{9}-\frac{24661}{103416}a^{8}-\frac{193663}{206832}a^{7}+\frac{87743}{51708}a^{6}+\frac{136643}{51708}a^{5}-\frac{20682}{4309}a^{4}-\frac{654115}{103416}a^{3}+\frac{2476}{12927}a^{2}+\frac{1663759}{103416}a-\frac{624931}{25854}$, $\frac{8183}{620496}a^{9}+\frac{317}{38781}a^{8}-\frac{16039}{103416}a^{7}-\frac{7543}{51708}a^{6}+\frac{23018}{38781}a^{5}+\frac{31502}{38781}a^{4}-\frac{20335}{51708}a^{3}-\frac{47867}{77562}a^{2}+\frac{418279}{155124}a+\frac{185554}{38781}$, $\frac{17233}{1240992}a^{9}-\frac{11789}{310248}a^{8}-\frac{26975}{206832}a^{7}+\frac{2174}{12927}a^{6}+\frac{67211}{155124}a^{5}-\frac{22457}{77562}a^{4}-\frac{56603}{103416}a^{3}-\frac{5321}{77562}a^{2}+\frac{620825}{310248}a-\frac{169327}{77562}$ | 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: | \( 4856.88200343 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 4856.88200343 \cdot 1}{2\cdot\sqrt{576000000000000}}\cr\approx \mathstrut & 0.630805948285 \end{aligned}\]
Galois group
$A_5^2:C_4$ (as 10T42):
A non-solvable group of order 14400 |
The 22 conjugacy class representatives for $A_5^2 : C_4$ |
Character table for $A_5^2 : C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 25 sibling: | data not computed |
Degree 30 sibling: | data not computed |
Degree 36 sibling: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.8.18.22 | $x^{8} + 8 x^{7} + 6 x^{6} - 24 x^{5} + 248 x^{4} - 32 x^{3} + 172 x^{2} + 96 x + 84$ | $4$ | $2$ | $18$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $[2, 2, 3, 7/2, 7/2]^{2}$ | |
\(3\) | 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(5\) | 5.10.12.15 | $x^{10} + 5 x^{3} + 5$ | $10$ | $1$ | $12$ | $(C_5^2 : C_8):C_2$ | $[11/8, 11/8]_{8}^{2}$ |