Normalized defining polynomial
\( x^{10} - 5x^{9} + 5x^{8} + 10x^{7} - 30x^{6} - 2x^{5} + 135x^{4} - 215x^{3} - 260x^{2} + 415x - 109 \)
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: | \(-3690562500000000\) \(\medspace = -\,2^{8}\cdot 3^{10}\cdot 5^{12}\) | 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: | \(36.03\) | 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: | $2^{15/8}3^{11/6}5^{271/200}\approx 243.36487591537934$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | 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: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{5}-\frac{1}{3}a^{2}$, $\frac{1}{140542851}a^{9}-\frac{10789798}{140542851}a^{8}-\frac{4780535}{46847617}a^{7}+\frac{7784557}{46847617}a^{6}+\frac{16125298}{140542851}a^{5}-\frac{14889658}{46847617}a^{4}-\frac{50275355}{140542851}a^{3}-\frac{64172120}{140542851}a^{2}+\frac{15703576}{46847617}a+\frac{5516975}{140542851}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $6$ | 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: | $\frac{343}{46847617}a^{9}+\frac{61029}{46847617}a^{8}-\frac{170730}{46847617}a^{7}-\frac{633354}{46847617}a^{6}+\frac{2958408}{46847617}a^{5}-\frac{2287323}{46847617}a^{4}-\frac{4523709}{46847617}a^{3}+\frac{7342830}{46847617}a^{2}-\frac{3448161}{46847617}a-\frac{75277489}{46847617}$, $\frac{5377736}{140542851}a^{9}-\frac{23130617}{140542851}a^{8}+\frac{6903671}{140542851}a^{7}+\frac{27633667}{46847617}a^{6}-\frac{154358692}{140542851}a^{5}-\frac{94186133}{140542851}a^{4}+\frac{739315013}{140542851}a^{3}-\frac{700689691}{140542851}a^{2}-\frac{2292612788}{140542851}a+\frac{1785193861}{140542851}$, $\frac{1084211}{140542851}a^{9}-\frac{5543074}{140542851}a^{8}+\frac{7202666}{140542851}a^{7}+\frac{13257187}{140542851}a^{6}-\frac{70956437}{140542851}a^{5}+\frac{98704150}{140542851}a^{4}-\frac{16635725}{140542851}a^{3}-\frac{286314536}{140542851}a^{2}+\frac{106395742}{140542851}a-\frac{42049101}{46847617}$, $\frac{7013897}{46847617}a^{9}-\frac{97344083}{140542851}a^{8}+\frac{68060848}{140542851}a^{7}+\frac{79896705}{46847617}a^{6}-\frac{541104088}{140542851}a^{5}-\frac{262267186}{140542851}a^{4}+\frac{911093201}{46847617}a^{3}-\frac{3444292387}{140542851}a^{2}-\frac{6756530356}{140542851}a+\frac{2033073744}{46847617}$, $\frac{159793}{46847617}a^{9}-\frac{1343363}{46847617}a^{8}+\frac{3640641}{46847617}a^{7}-\frac{9279415}{140542851}a^{6}-\frac{2886920}{46847617}a^{5}+\frac{7258972}{46847617}a^{4}+\frac{88246807}{140542851}a^{3}-\frac{108619349}{46847617}a^{2}+\frac{114678808}{46847617}a-\frac{107106827}{140542851}$, $\frac{102434507}{140542851}a^{9}-\frac{481620190}{140542851}a^{8}+\frac{397844848}{140542851}a^{7}+\frac{336894852}{46847617}a^{6}-\frac{2542535855}{140542851}a^{5}-\frac{1036354558}{140542851}a^{4}+\frac{13232002298}{140542851}a^{3}-\frac{18063589754}{140542851}a^{2}-\frac{29021401831}{140542851}a+\frac{28016155783}{140542851}$ (assuming GRH) | 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: | \( 19015.8120449 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 19015.8120449 \cdot 1}{2\cdot\sqrt{3690562500000000}}\cr\approx \mathstrut & 0.621152426208 \end{aligned}\] (assuming GRH)
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 | ${\href{/padicField/7.10.0.1}{10} }$ | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{5}$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.8.8.10 | $x^{8} - 6 x^{7} + 36 x^{6} - 60 x^{5} + 88 x^{4} + 136 x^{3} + 336 x^{2} + 400 x + 144$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ | |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.6.8.9 | $x^{6} + 36 x^{4} + 6 x^{3} + 9$ | $3$ | $2$ | $8$ | $S_3\times C_3$ | $[2, 2]^{2}$ | |
\(5\) | 5.10.12.15 | $x^{10} + 5 x^{3} + 5$ | $10$ | $1$ | $12$ | $(C_5^2 : C_8):C_2$ | $[11/8, 11/8]_{8}^{2}$ |