Normalized defining polynomial
\( x^{10} - 5x^{9} + 20x^{8} - 25x^{7} + 25x^{6} + 78x^{5} - 275x^{4} + 985x^{3} + 1490x^{2} - 625x - 449 \)
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: | \(14762250000000000\) \(\medspace = 2^{10}\cdot 3^{10}\cdot 5^{12}\) | 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: | \(41.39\) | 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: | $2^{2}3^{3/2}5^{271/200}\approx 184.0120354400766$ | ||
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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{22}a^{8}+\frac{1}{22}a^{7}-\frac{2}{11}a^{6}-\frac{1}{11}a^{5}-\frac{5}{11}a^{4}-\frac{5}{11}a^{3}+\frac{9}{22}a^{2}-\frac{3}{22}a-\frac{4}{11}$, $\frac{1}{38498869046}a^{9}+\frac{15617518}{1749948593}a^{8}-\frac{2289111599}{38498869046}a^{7}+\frac{5463920629}{38498869046}a^{6}-\frac{427270353}{19249434523}a^{5}-\frac{761229723}{1749948593}a^{4}-\frac{10312678797}{38498869046}a^{3}+\frac{9024554621}{19249434523}a^{2}-\frac{1203812219}{38498869046}a-\frac{17638912591}{38498869046}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (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: | $\frac{919911}{938996806}a^{9}-\frac{4265091}{938996806}a^{8}+\frac{16948977}{938996806}a^{7}-\frac{7674905}{469498403}a^{6}+\frac{5929368}{469498403}a^{5}+\frac{48660972}{469498403}a^{4}-\frac{216109909}{938996806}a^{3}+\frac{718696053}{938996806}a^{2}+\frac{1854249675}{938996806}a+\frac{29083211}{469498403}$, $\frac{919911}{938996806}a^{9}-\frac{4265091}{938996806}a^{8}+\frac{16948977}{938996806}a^{7}-\frac{7674905}{469498403}a^{6}+\frac{5929368}{469498403}a^{5}+\frac{48660972}{469498403}a^{4}-\frac{216109909}{938996806}a^{3}+\frac{718696053}{938996806}a^{2}+\frac{915252869}{938996806}a-\frac{440415192}{469498403}$, $\frac{603825537}{38498869046}a^{9}-\frac{3933817609}{38498869046}a^{8}+\frac{18364006883}{38498869046}a^{7}-\frac{21473211772}{19249434523}a^{6}+\frac{41416631356}{19249434523}a^{5}-\frac{34147164984}{19249434523}a^{4}-\frac{55400353729}{38498869046}a^{3}+\frac{715048611647}{38498869046}a^{2}-\frac{256985016027}{38498869046}a-\frac{119432574304}{19249434523}$, $\frac{7338416967}{19249434523}a^{9}-\frac{50793382792}{19249434523}a^{8}+\frac{240712111709}{19249434523}a^{7}-\frac{1229742510895}{38498869046}a^{6}+\frac{1209792390523}{19249434523}a^{5}-\frac{1353382712229}{19249434523}a^{4}-\frac{206011049754}{19249434523}a^{3}+\frac{8621543405711}{19249434523}a^{2}-\frac{5440223908113}{19249434523}a+\frac{18030674289}{3499897186}$, $\frac{260509180617}{38498869046}a^{9}-\frac{1423650320885}{38498869046}a^{8}+\frac{2931275231693}{19249434523}a^{7}-\frac{9195623938027}{38498869046}a^{6}+\frac{5293791769328}{19249434523}a^{5}+\frac{7793887189126}{19249434523}a^{4}-\frac{79456066630217}{38498869046}a^{3}+\frac{291818081012841}{38498869046}a^{2}+\frac{125987471127944}{19249434523}a-\frac{295500556882667}{38498869046}$ (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)]
| |
Regulator: | \( 44170.5306745 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 44170.5306745 \cdot 1}{2\cdot\sqrt{14762250000000000}}\cr\approx \mathstrut & 1.13319802298 \end{aligned}\] (assuming GRH)
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 | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{7}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
\(3\) | 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
3.6.8.7 | $x^{6} + 6 x^{5} + 9 x^{4} + 6 x^{3} + 18 x^{2} + 90$ | $3$ | $2$ | $8$ | $S_3\times C_3$ | $[2]^{6}$ | |
\(5\) | 5.10.12.15 | $x^{10} + 5 x^{3} + 5$ | $10$ | $1$ | $12$ | $(C_5^2 : C_8):C_2$ | $[11/8, 11/8]_{8}^{2}$ |