Normalized defining polynomial
\( x^{10} + 4x^{8} - 68x^{7} - 49x^{6} - 262x^{5} + 1050x^{4} + 1550x^{3} + 5925x^{2} + 10849x + 18989 \)
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: | \(-746534197277346875\) \(\medspace = -\,5^{5}\cdot 751^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(61.28\) | 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: | $5^{1/2}751^{1/2}\approx 61.27805479941412$ | ||
Ramified primes: | \(5\), \(751\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3755}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{27}a^{8}-\frac{1}{27}a^{7}-\frac{2}{27}a^{6}+\frac{4}{27}a^{5}-\frac{1}{9}a^{4}+\frac{10}{27}a^{3}-\frac{10}{27}a^{2}-\frac{4}{27}a-\frac{4}{27}$, $\frac{1}{13904978669001}a^{9}-\frac{154844566175}{13904978669001}a^{8}+\frac{1839474491975}{13904978669001}a^{7}+\frac{46790332952}{4634992889667}a^{6}-\frac{5337395426068}{13904978669001}a^{5}+\frac{5403975279961}{13904978669001}a^{4}-\frac{1787072093417}{13904978669001}a^{3}+\frac{2117816114759}{4634992889667}a^{2}-\frac{131918462315}{1544997629889}a+\frac{6707958202093}{13904978669001}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
$C_{20}$, which has order $20$
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: | $\frac{3250172722}{13904978669001}a^{9}-\frac{8089929836}{13904978669001}a^{8}+\frac{11922865583}{13904978669001}a^{7}-\frac{82178611654}{4634992889667}a^{6}+\frac{286846596098}{13904978669001}a^{5}-\frac{380715776615}{13904978669001}a^{4}+\frac{4809025833670}{13904978669001}a^{3}-\frac{172840164919}{4634992889667}a^{2}+\frac{313666650116}{514999209963}a+\frac{5366076767008}{13904978669001}$, $\frac{145599154}{4634992889667}a^{9}-\frac{2206893386}{4634992889667}a^{8}-\frac{9212188411}{4634992889667}a^{7}+\frac{1936079329}{514999209963}a^{6}+\frac{120981736961}{4634992889667}a^{5}+\frac{585363220588}{4634992889667}a^{4}-\frac{318725054363}{4634992889667}a^{3}-\frac{1365967633847}{1544997629889}a^{2}-\frac{2621564211856}{1544997629889}a-\frac{4767945940064}{4634992889667}$, $\frac{5514952094}{1544997629889}a^{9}+\frac{81714643009}{4634992889667}a^{8}+\frac{404104970342}{4634992889667}a^{7}+\frac{581410844845}{4634992889667}a^{6}+\frac{535643039221}{4634992889667}a^{5}-\frac{1441124295794}{1544997629889}a^{4}-\frac{9809507679824}{4634992889667}a^{3}-\frac{23400573973381}{4634992889667}a^{2}-\frac{34393678814443}{4634992889667}a-\frac{44993591337454}{4634992889667}$, $\frac{263535256142}{13904978669001}a^{9}-\frac{792616775908}{13904978669001}a^{8}+\frac{1836774664099}{13904978669001}a^{7}-\frac{4148800804810}{4634992889667}a^{6}+\frac{18581839568191}{13904978669001}a^{5}-\frac{6744985251043}{13904978669001}a^{4}+\frac{40637567350022}{13904978669001}a^{3}+\frac{5375768816878}{1544997629889}a^{2}+\frac{116275087786325}{4634992889667}a-\frac{185543199186145}{13904978669001}$ | 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: | \( 12177.78378170143 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 12177.78378170143 \cdot 20}{2\cdot\sqrt{746534197277346875}}\cr\approx \mathstrut & 1.38020230151780 \end{aligned}\]
Galois group
A solvable group of order 20 |
The 8 conjugacy class representatives for $D_{10}$ |
Character table for $D_{10}$ |
Intermediate fields
\(\Q(\sqrt{-3755}) \), 5.1.564001.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 20.0.557313307704532662314090039072265625.1 |
Degree 10 sibling: | 10.2.994053525003125.2 |
Minimal sibling: | 10.2.994053525003125.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.10.0.1}{10} }$ | ${\href{/padicField/3.2.0.1}{2} }^{4}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{5}$ | ${\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.1.0.1}{1} }^{10}$ | ${\href{/padicField/23.10.0.1}{10} }$ | ${\href{/padicField/29.2.0.1}{2} }^{5}$ | ${\href{/padicField/31.2.0.1}{2} }^{5}$ | ${\href{/padicField/37.10.0.1}{10} }$ | ${\href{/padicField/41.2.0.1}{2} }^{5}$ | ${\href{/padicField/43.10.0.1}{10} }$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.2.0.1}{2} }^{5}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.10.5.1 | $x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
\(751\) | 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 | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.751.2t1.a.a | $1$ | $ 751 $ | \(\Q(\sqrt{-751}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.3755.2t1.a.a | $1$ | $ 5 \cdot 751 $ | \(\Q(\sqrt{-3755}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 2.18775.10t3.a.b | $2$ | $ 5^{2} \cdot 751 $ | 10.0.746534197277346875.2 | $D_{10}$ (as 10T3) | $1$ | $0$ |
* | 2.751.5t2.a.a | $2$ | $ 751 $ | 5.1.564001.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.751.5t2.a.b | $2$ | $ 751 $ | 5.1.564001.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.18775.10t3.a.a | $2$ | $ 5^{2} \cdot 751 $ | 10.0.746534197277346875.2 | $D_{10}$ (as 10T3) | $1$ | $0$ |