Normalized defining polynomial
\( x^{10} - 2x^{9} + 3x^{8} + x^{7} - 15x^{6} + 22x^{5} + 9x^{4} - 65x^{3} + 77x^{2} - 39x + 9 \)
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: | \(-11592740743\) \(\medspace = -\,103^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(10.15\) | 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: | $103^{1/2}\approx 10.14889156509222$ | ||
Ramified primes: | \(103\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-103}) \) | ||
$\card{ \Gal(K/\Q) }$: | $10$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{15}a^{8}-\frac{1}{15}a^{7}-\frac{1}{3}a^{6}-\frac{7}{15}a^{5}-\frac{2}{15}a^{4}-\frac{1}{15}a^{3}+\frac{7}{15}a^{2}-\frac{1}{15}a-\frac{1}{5}$, $\frac{1}{14115}a^{9}-\frac{31}{14115}a^{8}-\frac{196}{2823}a^{7}-\frac{2632}{14115}a^{6}+\frac{1033}{14115}a^{5}+\frac{6764}{14115}a^{4}+\frac{5227}{14115}a^{3}+\frac{794}{14115}a^{2}+\frac{2074}{4705}a+\frac{13}{941}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{566}{14115}a^{9}+\frac{443}{2823}a^{8}-\frac{431}{14115}a^{7}+\frac{6478}{14115}a^{6}+\frac{4081}{14115}a^{5}-\frac{22148}{14115}a^{4}+\frac{26326}{14115}a^{3}+\frac{37246}{14115}a^{2}-\frac{88024}{14115}a+\frac{12324}{4705}$, $\frac{2864}{14115}a^{9}-\frac{1051}{4705}a^{8}+\frac{5929}{14115}a^{7}+\frac{2924}{4705}a^{6}-\frac{7150}{2823}a^{5}+\frac{10888}{4705}a^{4}+\frac{54337}{14115}a^{3}-\frac{49059}{4705}a^{2}+\frac{95132}{14115}a-\frac{7686}{4705}$, $\frac{2864}{14115}a^{9}-\frac{1051}{4705}a^{8}+\frac{5929}{14115}a^{7}+\frac{2924}{4705}a^{6}-\frac{7150}{2823}a^{5}+\frac{10888}{4705}a^{4}+\frac{54337}{14115}a^{3}-\frac{49059}{4705}a^{2}+\frac{95132}{14115}a-\frac{2981}{4705}$, $\frac{1747}{4705}a^{9}-\frac{5324}{14115}a^{8}+\frac{9208}{14115}a^{7}+\frac{14863}{14115}a^{6}-\frac{66431}{14115}a^{5}+\frac{9188}{2823}a^{4}+\frac{20765}{2823}a^{3}-\frac{243467}{14115}a^{2}+\frac{27683}{2823}a-\frac{9387}{4705}$ | 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: | \( 6.90703427277 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 6.90703427277 \cdot 1}{2\cdot\sqrt{11592740743}}\cr\approx \mathstrut & 0.314099970311 \end{aligned}\]
Galois group
A solvable group of order 10 |
The 4 conjugacy class representatives for $D_5$ |
Character table for $D_5$ |
Intermediate fields
\(\Q(\sqrt{-103}) \), 5.1.10609.1 x5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.1.10609.1 |
Minimal sibling: | 5.1.10609.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{5}$ | ${\href{/padicField/5.2.0.1}{2} }^{5}$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{5}$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{5}$ | ${\href{/padicField/37.2.0.1}{2} }^{5}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}$ | ${\href{/padicField/53.2.0.1}{2} }^{5}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(103\) | 103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |