Normalized defining polynomial
\( x^{10} - 4x^{9} + 26x^{8} - 78x^{7} + 254x^{6} - 496x^{5} + 988x^{4} - 1017x^{3} + 1219x^{2} - 935x + 1211 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-2436910203746875\) \(\medspace = -\,5^{5}\cdot 239^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(34.57\) | 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: | $5^{1/2}239^{1/2}\approx 34.56877203488721$ | ||
Ramified primes: | \(5\), \(239\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1195}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{374769890761}a^{9}-\frac{39468406718}{374769890761}a^{8}-\frac{89674317289}{374769890761}a^{7}+\frac{13654474119}{53538555823}a^{6}+\frac{5484121001}{374769890761}a^{5}+\frac{55738360066}{374769890761}a^{4}-\frac{12537569637}{53538555823}a^{3}+\frac{113776534065}{374769890761}a^{2}+\frac{122819119176}{374769890761}a+\frac{26252663619}{53538555823}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{8}$, which has order $8$
Unit group
Rank: | $4$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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)
| |
Fundamental units: | $\frac{94176088}{374769890761}a^{9}-\frac{320166071}{374769890761}a^{8}+\frac{3487035346}{374769890761}a^{7}-\frac{1903104761}{53538555823}a^{6}+\frac{52145854661}{374769890761}a^{5}-\frac{143480787629}{374769890761}a^{4}+\frac{48680201944}{53538555823}a^{3}-\frac{572432119796}{374769890761}a^{2}+\frac{598713996540}{374769890761}a-\frac{85853944720}{53538555823}$, $\frac{450446196}{374769890761}a^{9}-\frac{1970995729}{374769890761}a^{8}+\frac{10133053054}{374769890761}a^{7}-\frac{4755499191}{53538555823}a^{6}+\frac{86804669281}{374769890761}a^{5}-\frac{172433195646}{374769890761}a^{4}+\frac{37354334345}{53538555823}a^{3}-\frac{212209315473}{374769890761}a^{2}+\frac{52214220953}{374769890761}a-\frac{3814512140}{53538555823}$, $\frac{386887936}{374769890761}a^{9}+\frac{3074758835}{374769890761}a^{8}-\frac{13520915321}{374769890761}a^{7}+\frac{15061075569}{53538555823}a^{6}-\frac{368427828753}{374769890761}a^{5}+\frac{1213921099780}{374769890761}a^{4}-\frac{378294750460}{53538555823}a^{3}+\frac{4887246677717}{374769890761}a^{2}-\frac{5328078027148}{374769890761}a+\frac{387268675417}{53538555823}$, $\frac{2017949047}{374769890761}a^{9}-\frac{8497091090}{374769890761}a^{8}+\frac{45532036395}{374769890761}a^{7}-\frac{17162254418}{53538555823}a^{6}+\frac{327457141470}{374769890761}a^{5}-\frac{532524803544}{374769890761}a^{4}+\frac{133800356327}{53538555823}a^{3}-\frac{832200055348}{374769890761}a^{2}+\frac{774854829220}{374769890761}a-\frac{86080339203}{53538555823}$ | 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: | \( 430.654786243 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 430.654786243 \cdot 8}{2\cdot\sqrt{2436910203746875}}\cr\approx \mathstrut & 0.341718780787 \end{aligned}\]
Galois group
A solvable group of order 20 |
The 8 conjugacy class representatives for $D_{10}$ |
Character table for $D_{10}$ |
Intermediate fields
\(\Q(\sqrt{-1195}) \), 5.1.57121.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 20.0.5938531341125635825289072265625.1 |
Degree 10 sibling: | 10.2.10196277003125.1 |
Minimal sibling: | 10.2.10196277003125.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.10.0.1}{10} }$ | ${\href{/padicField/3.10.0.1}{10} }$ | R | ${\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{5}$ | ${\href{/padicField/19.2.0.1}{2} }^{5}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{5}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{5}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.10.5.1 | $x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
\(239\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.239.2t1.a.a | $1$ | $ 239 $ | \(\Q(\sqrt{-239}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.1195.2t1.a.a | $1$ | $ 5 \cdot 239 $ | \(\Q(\sqrt{-1195}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 2.5975.10t3.a.a | $2$ | $ 5^{2} \cdot 239 $ | 10.0.2436910203746875.1 | $D_{10}$ (as 10T3) | $1$ | $0$ |
* | 2.239.5t2.a.b | $2$ | $ 239 $ | 5.1.57121.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.239.5t2.a.a | $2$ | $ 239 $ | 5.1.57121.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.5975.10t3.a.b | $2$ | $ 5^{2} \cdot 239 $ | 10.0.2436910203746875.1 | $D_{10}$ (as 10T3) | $1$ | $0$ |