Normalized defining polynomial
\( x^{10} - 2x^{9} - 12x^{7} + 33x^{6} + 21x^{4} - 60x^{3} + 216x^{2} - 154x + 101 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-2975389355520000000\) \(\medspace = -\,2^{15}\cdot 3^{19}\cdot 5^{7}\) | 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: | \(70.36\) | 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: | $2^{9/4}3^{53/24}5^{7/8}\approx 220.06907340902904$ | ||
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: | \(\Q(\sqrt{-30}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6}a^{6}-\frac{1}{6}a^{5}-\frac{1}{6}a^{4}+\frac{1}{6}a^{2}+\frac{1}{6}a-\frac{1}{6}$, $\frac{1}{6}a^{7}-\frac{1}{3}a^{5}-\frac{1}{6}a^{4}+\frac{1}{6}a^{3}+\frac{1}{3}a^{2}-\frac{1}{6}$, $\frac{1}{6}a^{8}-\frac{1}{2}a^{5}-\frac{1}{6}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{6}a-\frac{1}{3}$, $\frac{1}{34980}a^{9}-\frac{103}{3180}a^{8}-\frac{107}{3180}a^{7}+\frac{129}{2332}a^{6}+\frac{907}{8745}a^{5}+\frac{263}{8745}a^{4}-\frac{157}{11660}a^{3}-\frac{3719}{34980}a^{2}-\frac{2903}{6996}a+\frac{10691}{34980}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $4$ | 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: | $\frac{14781}{11660}a^{9}-\frac{7739}{3180}a^{8}-\frac{23461}{3180}a^{7}+\frac{34831}{2332}a^{6}+\frac{231977}{17490}a^{5}-\frac{98609}{5830}a^{4}-\frac{270673}{34980}a^{3}+\frac{1219001}{11660}a^{2}-\frac{604151}{6996}a+\frac{1899313}{34980}$, $\frac{34090}{1749}a^{9}-\frac{9995}{106}a^{8}+\frac{64903}{318}a^{7}-\frac{296542}{583}a^{6}+\frac{1674393}{1166}a^{5}-\frac{4312433}{1749}a^{4}+\frac{3385101}{1166}a^{3}-\frac{1081975}{583}a^{2}+\frac{2795957}{3498}a-\frac{181007}{3498}$, $\frac{243847}{6996}a^{9}+\frac{10741}{636}a^{8}+\frac{12673}{636}a^{7}-\frac{996189}{2332}a^{6}-\frac{219757}{3498}a^{5}-\frac{410420}{1749}a^{4}+\frac{107501}{6996}a^{3}-\frac{5174319}{2332}a^{2}+\frac{3571603}{2332}a-\frac{2994735}{2332}$, $\frac{443942375}{2332}a^{9}-\frac{60669435}{212}a^{8}-\frac{51272583}{212}a^{7}-\frac{15944353933}{6996}a^{6}+\frac{18565461539}{3498}a^{5}+\frac{6765846580}{1749}a^{4}+\frac{8024133133}{2332}a^{3}-\frac{86363435611}{6996}a^{2}+\frac{227668102727}{6996}a-\frac{50962928315}{6996}$ (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: | \( 1229530.9813 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 1229530.9813 \cdot 1}{2\cdot\sqrt{2975389355520000000}}\cr\approx \mathstrut & 3.4900923600 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 1440 |
The 13 conjugacy class representatives for $(A_6 : C_2) : C_2$ |
Character table for $(A_6 : C_2) : C_2$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 12 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 30 sibling: | data not computed |
Degree 36 sibling: | data not computed |
Degree 40 siblings: | data not computed |
Degree 45 sibling: | 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 | ${\href{/padicField/7.10.0.1}{10} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.3.0.1}{3} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
2.4.6.8 | $x^{4} + 2 x^{3} + 2$ | $4$ | $1$ | $6$ | $D_{4}$ | $[2, 2]^{2}$ | |
2.4.6.5 | $x^{4} - 4 x^{3} + 36 x^{2} + 8 x + 148$ | $2$ | $2$ | $6$ | $D_{4}$ | $[2, 3]^{2}$ | |
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.9.19.57 | $x^{9} + 18 x^{3} + 18 x^{2} + 21$ | $9$ | $1$ | $19$ | $(C_3^2:C_8):C_2$ | $[19/8, 19/8]_{8}^{2}$ | |
\(5\) | 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
5.8.7.3 | $x^{8} + 10$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |