Normalized defining polynomial
\( x^{10} - 5x^{9} - 5x^{8} + 50x^{7} - 5x^{6} - 145x^{5} - 45x^{4} + 190x^{3} + 175x^{2} - 25x - 55 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-2304000000000000\) \(\medspace = -\,2^{20}\cdot 3^{2}\cdot 5^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(34.38\) | 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))
| |
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: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{245006}a^{9}+\frac{55339}{245006}a^{8}-\frac{7946}{122503}a^{7}+\frac{22371}{122503}a^{6}-\frac{74399}{245006}a^{5}+\frac{32435}{245006}a^{4}+\frac{23068}{122503}a^{3}-\frac{50779}{122503}a^{2}+\frac{56869}{245006}a+\frac{10835}{245006}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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: | $\frac{9188}{122503}a^{9}-\frac{55221}{122503}a^{8}+\frac{7880}{122503}a^{7}+\frac{459440}{122503}a^{6}-\frac{501284}{122503}a^{5}-\frac{894540}{122503}a^{4}+\frac{527200}{122503}a^{3}+\frac{1337980}{122503}a^{2}+\frac{159580}{122503}a-\frac{655474}{122503}$, $\frac{93}{122503}a^{9}+\frac{1401}{122503}a^{8}-\frac{7920}{122503}a^{7}-\frac{4096}{122503}a^{6}+\frac{63564}{122503}a^{5}-\frac{46120}{122503}a^{4}-\frac{119460}{122503}a^{3}+\frac{110340}{122503}a^{2}+\frac{143691}{122503}a+\frac{27631}{122503}$, $\frac{41501}{245006}a^{9}-\frac{92454}{122503}a^{8}-\frac{111373}{122503}a^{7}+\frac{826155}{122503}a^{6}-\frac{67287}{245006}a^{5}-\frac{1642805}{122503}a^{4}-\frac{505889}{122503}a^{3}+\frac{1142357}{122503}a^{2}+\frac{957595}{245006}a-\frac{390098}{122503}$, $\frac{7266}{122503}a^{9}-\frac{45847}{245006}a^{8}-\frac{73446}{122503}a^{7}+\frac{217416}{122503}a^{6}+\frac{267611}{122503}a^{5}-\frac{658639}{245006}a^{4}-\frac{556547}{122503}a^{3}-\frac{574871}{122503}a^{2}+\frac{252541}{122503}a+\frac{1017889}{245006}$, $\frac{98005}{245006}a^{9}-\frac{283318}{122503}a^{8}+\frac{3841}{122503}a^{7}+\frac{2361221}{122503}a^{6}-\frac{4750549}{245006}a^{5}-\frac{4429194}{122503}a^{4}+\frac{2436535}{122503}a^{3}+\frac{5846121}{122503}a^{2}+\frac{2499917}{245006}a-\frac{2497226}{122503}$, $\frac{2048}{122503}a^{9}+\frac{18997}{122503}a^{8}-\frac{206024}{122503}a^{7}+\frac{244378}{122503}a^{6}+\frac{1494616}{122503}a^{5}-\frac{3154824}{122503}a^{4}-\frac{1678327}{122503}a^{3}+\frac{5286939}{122503}a^{2}+\frac{824880}{122503}a-\frac{1820508}{122503}$ | 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: | \( 12790.700327 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 12790.700327 \cdot 1}{2\cdot\sqrt{2304000000000000}}\cr\approx \mathstrut & 0.52878932436 \end{aligned}\]
Galois group
$S_5\wr C_2$ (as 10T43):
A non-solvable group of order 28800 |
The 35 conjugacy class representatives for $S_5^2 \wr C_2$ |
Character table for $S_5^2 \wr C_2$ |
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 siblings: | 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 | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.8.20.40 | $x^{8} + 4 x^{7} + 28 x^{6} + 40 x^{5} + 84 x^{4} + 8 x^{3} + 88 x^{2} - 16 x + 4$ | $4$ | $2$ | $20$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $[2, 2, 3, 7/2, 7/2]^{2}$ | |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.6.0.1 | $x^{6} + 2 x^{4} + x^{2} + 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(5\) | 5.10.12.15 | $x^{10} + 5 x^{3} + 5$ | $10$ | $1$ | $12$ | $(C_5^2 : C_8):C_2$ | $[11/8, 11/8]_{8}^{2}$ |