Normalized defining polynomial
\( x^{10} - x^{9} + 4x^{8} - 17x^{7} + 60x^{6} + 11x^{5} - 12x^{4} + 72x^{3} + 351x^{2} - 243x - 243 \)
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: | \(14461892424733\) \(\medspace = 13^{5}\cdot 79^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(20.70\) | 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: | $13^{1/2}79^{1/2}\approx 32.046840717924134$ | ||
Ramified primes: | \(13\), \(79\) | 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: | \(\Q(\sqrt{13}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{9}a^{7}-\frac{1}{9}a^{6}+\frac{4}{9}a^{5}+\frac{1}{9}a^{4}-\frac{1}{3}a^{3}+\frac{2}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{27}a^{8}-\frac{1}{27}a^{7}+\frac{4}{27}a^{6}+\frac{10}{27}a^{5}+\frac{2}{9}a^{4}+\frac{11}{27}a^{3}-\frac{4}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{4993284123}a^{9}-\frac{30339040}{4993284123}a^{8}-\frac{238976480}{4993284123}a^{7}+\frac{513466369}{4993284123}a^{6}-\frac{770434874}{1664428041}a^{5}+\frac{1025156918}{4993284123}a^{4}+\frac{68538139}{554809347}a^{3}+\frac{69072736}{184936449}a^{2}-\frac{87082253}{184936449}a-\frac{22978149}{61645483}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$
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)
| |
Fundamental units: | $\frac{8483848}{4993284123}a^{9}-\frac{28154455}{4993284123}a^{8}+\frac{34719958}{4993284123}a^{7}-\frac{215965577}{4993284123}a^{6}+\frac{258739295}{1664428041}a^{5}-\frac{740941627}{4993284123}a^{4}-\frac{375277978}{1664428041}a^{3}-\frac{15699589}{554809347}a^{2}+\frac{35592310}{184936449}a-\frac{71409377}{61645483}$, $\frac{233161}{184936449}a^{9}-\frac{28569}{61645483}a^{8}+\frac{276598}{61645483}a^{7}-\frac{1423755}{61645483}a^{6}+\frac{3323667}{61645483}a^{5}+\frac{4066587}{61645483}a^{4}+\frac{2471418}{61645483}a^{3}+\frac{7454673}{61645483}a^{2}-\frac{28085206}{184936449}a+\frac{19670607}{61645483}$, $\frac{242847}{61645483}a^{9}-\frac{9686}{61645483}a^{8}+\frac{885681}{61645483}a^{7}-\frac{3298605}{61645483}a^{6}+\frac{10299555}{61645483}a^{5}+\frac{12642318}{61645483}a^{4}+\frac{9285597}{61645483}a^{3}+\frac{24899238}{61645483}a^{2}+\frac{107603316}{61645483}a+\frac{36193939}{61645483}$, $\frac{2469352}{184936449}a^{9}+\frac{781346}{1664428041}a^{8}+\frac{17808481}{1664428041}a^{7}-\frac{433419700}{1664428041}a^{6}+\frac{640734305}{1664428041}a^{5}+\frac{633835496}{554809347}a^{4}+\frac{301510687}{1664428041}a^{3}-\frac{356049161}{184936449}a^{2}+\frac{36690071}{61645483}a+\frac{99999652}{61645483}$, $\frac{118032197}{554809347}a^{9}+\frac{11311439}{184936449}a^{8}+\frac{459098261}{554809347}a^{7}-\frac{1508871883}{554809347}a^{6}+\frac{4891442185}{554809347}a^{5}+\frac{2759323672}{184936449}a^{4}+\frac{8205614695}{554809347}a^{3}+\frac{12831017779}{554809347}a^{2}+\frac{15062097598}{184936449}a+\frac{2532061898}{61645483}$ | 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: | \( 835.981549877 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 835.981549877 \cdot 1}{2\cdot\sqrt{14461892424733}}\cr\approx \mathstrut & 0.685225629328 \end{aligned}\]
Galois group
A solvable group of order 20 |
The 8 conjugacy class representatives for $D_{10}$ |
Character table for $D_{10}$ |
Intermediate fields
\(\Q(\sqrt{13}) \), 5.1.6241.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 20.0.1305282261160894865367626964649.1 |
Degree 10 sibling: | 10.0.1142489501553907.1 |
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 | ${\href{/padicField/2.10.0.1}{10} }$ | ${\href{/padicField/3.2.0.1}{2} }^{4}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.10.0.1}{10} }$ | ${\href{/padicField/7.2.0.1}{2} }^{5}$ | ${\href{/padicField/11.10.0.1}{10} }$ | R | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }$ | ${\href{/padicField/37.2.0.1}{2} }^{5}$ | ${\href{/padicField/41.2.0.1}{2} }^{5}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{5}$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.10.5.1 | $x^{10} + 650 x^{9} + 169065 x^{8} + 22003800 x^{7} + 1434642698 x^{6} + 37701182242 x^{5} + 18651037600 x^{4} + 3808243140 x^{3} + 6315953361 x^{2} + 164195122608 x + 421659070668$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
\(79\) | $\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.79.2t1.a.a | $1$ | $ 79 $ | \(\Q(\sqrt{-79}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.13.2t1.a.a | $1$ | $ 13 $ | \(\Q(\sqrt{13}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.1027.2t1.a.a | $1$ | $ 13 \cdot 79 $ | \(\Q(\sqrt{-1027}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.13351.10t3.a.b | $2$ | $ 13^{2} \cdot 79 $ | 10.2.14461892424733.1 | $D_{10}$ (as 10T3) | $1$ | $0$ |
* | 2.79.5t2.a.a | $2$ | $ 79 $ | 5.1.6241.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.79.5t2.a.b | $2$ | $ 79 $ | 5.1.6241.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.13351.10t3.a.a | $2$ | $ 13^{2} \cdot 79 $ | 10.2.14461892424733.1 | $D_{10}$ (as 10T3) | $1$ | $0$ |