Normalized defining polynomial
\( x^{10} - 5 x^{9} + 41 x^{8} - 134 x^{7} + 1082 x^{6} - 2798 x^{5} + 7844 x^{4} - 11171 x^{3} + \cdots + 52883 \)
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: | \(2045626201320218209573\) \(\medspace = 17^{9}\cdot 29^{7}\) | 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: | \(135.23\) | 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: | $17^{9/10}29^{5/6}\approx 211.86978578607543$ | ||
Ramified primes: | \(17\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{493}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{7594340124}a^{8}-\frac{1}{1898585031}a^{7}-\frac{777428926}{1898585031}a^{6}+\frac{867403501}{3797170062}a^{5}-\frac{233980270}{632861677}a^{4}-\frac{194892662}{632861677}a^{3}-\frac{70416787}{271226433}a^{2}+\frac{299571267}{2531446708}a+\frac{1527811651}{7594340124}$, $\frac{1}{827783073516}a^{9}+\frac{25}{413891536758}a^{8}-\frac{93808095499}{206945768379}a^{7}-\frac{48920389949}{413891536758}a^{6}-\frac{17741815841}{68981922793}a^{5}-\frac{15361273950}{68981922793}a^{4}-\frac{47254209001}{206945768379}a^{3}+\frac{86318952603}{275927691172}a^{2}+\frac{148784778517}{827783073516}a+\frac{68176409081}{137963845586}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$ (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: | $\frac{15140}{10162833}a^{9}-\frac{22710}{3387611}a^{8}+\frac{579590}{10162833}a^{7}-\frac{1710625}{10162833}a^{6}+\frac{15336235}{10162833}a^{5}-\frac{11407665}{3387611}a^{4}+\frac{95409955}{10162833}a^{3}-\frac{109701830}{10162833}a^{2}+\frac{3652321805}{10162833}a-\frac{1244942341}{10162833}$, $\frac{75663401}{19709120798}a^{9}-\frac{1578676388}{68981922793}a^{8}+\frac{12519642374}{68981922793}a^{7}-\frac{44651594070}{68981922793}a^{6}+\frac{298905258285}{68981922793}a^{5}-\frac{756706820971}{68981922793}a^{4}+\frac{1453267758485}{68981922793}a^{3}+\frac{888208618949}{19709120798}a^{2}+\frac{62631411931519}{137963845586}a-\frac{11170803311441}{68981922793}$, $\frac{266945792}{68981922793}a^{9}-\frac{1487750574}{68981922793}a^{8}+\frac{12719319608}{68981922793}a^{7}+\frac{8989032896}{68981922793}a^{6}-\frac{33201626268}{68981922793}a^{5}+\frac{860697204970}{68981922793}a^{4}-\frac{7781952315512}{68981922793}a^{3}+\frac{49175155862902}{68981922793}a^{2}-\frac{6218051163366}{9854560399}a+\frac{9368024360615}{68981922793}$, $\frac{208733317}{68981922793}a^{9}+\frac{3126421381}{137963845586}a^{8}-\frac{19482541851}{68981922793}a^{7}+\frac{137749566571}{68981922793}a^{6}-\frac{197387492678}{68981922793}a^{5}+\frac{61248452148}{68981922793}a^{4}-\frac{2708988106525}{68981922793}a^{3}+\frac{32101452968849}{68981922793}a^{2}-\frac{8569856162017}{19709120798}a+\frac{13248936769875}{137963845586}$, $\frac{13066218729}{68981922793}a^{9}-\frac{1336522924757}{275927691172}a^{8}-\frac{359092403023}{68981922793}a^{7}-\frac{12619770183492}{68981922793}a^{6}-\frac{44975060711963}{137963845586}a^{5}-\frac{342420512467623}{68981922793}a^{4}-\frac{940379332046533}{68981922793}a^{3}-\frac{30\!\cdots\!88}{68981922793}a^{2}+\frac{15\!\cdots\!11}{275927691172}a-\frac{38\!\cdots\!67}{275927691172}$ (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: | \( 4857519.38895 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 4857519.38895 \cdot 2}{2\cdot\sqrt{2045626201320218209573}}\cr\approx \mathstrut & 0.669546692007 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times S_5$ (as 10T22):
A non-solvable group of order 240 |
The 14 conjugacy class representatives for $S_5\times C_2$ |
Character table for $S_5\times C_2$ |
Intermediate fields
\(\Q(\sqrt{493}) \), 5.1.2036993669.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 10 sibling: | data not computed |
Degree 12 siblings: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 30 siblings: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | 10.2.70538834528283386537.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }^{2}$ | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.2.0.1}{2} }^{5}$ | R | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(17\) | 17.10.9.2 | $x^{10} + 51$ | $10$ | $1$ | $9$ | $F_{5}\times C_2$ | $[\ ]_{10}^{4}$ |
\(29\) | 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
29.6.5.2 | $x^{6} + 58$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ |