Normalized defining polynomial
\( x^{10} - 5x^{9} + 15x^{8} - 15x^{7} + 60x^{4} - 60x^{3} - 60x^{2} - 20x + 64 \)
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: | \(8303765625000000\) \(\medspace = 2^{6}\cdot 3^{12}\cdot 5^{12}\) | 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: | \(39.08\) | 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: | $2^{2/3}3^{25/18}5^{13/10}\approx 59.158353577804604$ | ||
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{44}a^{8}+\frac{1}{22}a^{7}-\frac{2}{11}a^{6}-\frac{1}{22}a^{5}+\frac{7}{44}a^{4}+\frac{1}{22}a^{3}-\frac{5}{11}a^{2}-\frac{5}{22}a+\frac{4}{11}$, $\frac{1}{2904}a^{9}+\frac{5}{968}a^{8}+\frac{105}{968}a^{7}-\frac{83}{968}a^{6}+\frac{17}{484}a^{5}-\frac{23}{484}a^{4}-\frac{87}{484}a^{3}-\frac{14}{121}a^{2}+\frac{20}{121}a-\frac{73}{363}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{4}$, which has order $4$ (assuming GRH)
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: | $\frac{53}{2904}a^{9}-\frac{131}{968}a^{8}+\frac{417}{968}a^{7}-\frac{747}{968}a^{6}+\frac{87}{484}a^{5}+\frac{57}{484}a^{4}-\frac{167}{484}a^{3}-\frac{115}{121}a^{2}+\frac{103}{121}a+\frac{289}{363}$, $\frac{43}{2904}a^{9}-\frac{49}{968}a^{8}+\frac{115}{968}a^{7}-\frac{5}{968}a^{6}+\frac{27}{484}a^{5}-\frac{219}{484}a^{4}+\frac{351}{484}a^{3}+\frac{58}{121}a^{2}-\frac{20}{121}a-\frac{367}{363}$, $\frac{35}{264}a^{9}-\frac{53}{88}a^{8}+\frac{139}{88}a^{7}-\frac{25}{88}a^{6}-\frac{167}{44}a^{5}+\frac{245}{44}a^{4}+\frac{181}{44}a^{3}-\frac{63}{11}a^{2}-\frac{115}{11}a+\frac{37}{33}$, $\frac{57}{968}a^{9}+\frac{151}{968}a^{8}-\frac{151}{968}a^{7}+\frac{1361}{968}a^{6}+\frac{777}{242}a^{5}+\frac{4}{121}a^{4}-\frac{3481}{484}a^{3}-\frac{2115}{242}a^{2}-\frac{717}{242}a-\frac{3}{121}$, $\frac{4141}{2904}a^{9}-\frac{5761}{968}a^{8}+\frac{16453}{968}a^{7}-\frac{9523}{968}a^{6}-\frac{179}{484}a^{5}-\frac{928}{121}a^{4}+\frac{38703}{484}a^{3}-\frac{1247}{121}a^{2}-\frac{19149}{242}a-\frac{37831}{363}$ (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: | \( 29752.5017991 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 29752.5017991 \cdot 4}{2\cdot\sqrt{8303765625000000}}\cr\approx \mathstrut & 4.07094663373 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 360 |
The 7 conjugacy class representatives for $\PSL(2,9)$ |
Character table for $\PSL(2,9)$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 6 siblings: | 6.2.182250000.1, 6.2.45562500.1 |
Degree 15 siblings: | data not computed |
Degree 20 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 36 sibling: | data not computed |
Degree 40 sibling: | data not computed |
Degree 45 sibling: | data not computed |
Minimal sibling: | 6.2.45562500.1 |
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 | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.9.12.21 | $x^{9} + 6 x^{4} + 3$ | $9$ | $1$ | $12$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ | |
\(5\) | 5.5.6.1 | $x^{5} + 10 x^{2} + 5$ | $5$ | $1$ | $6$ | $D_{5}$ | $[3/2]_{2}$ |
5.5.6.1 | $x^{5} + 10 x^{2} + 5$ | $5$ | $1$ | $6$ | $D_{5}$ | $[3/2]_{2}$ |