Normalized defining polynomial
\( x^{4} - x^{3} + 211x^{2} + 1032x + 17628 \)
Invariants
Degree: | $4$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(344454777\) \(\medspace = 3^{2}\cdot 337^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(136.23\) | 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: | $3^{1/2}337^{3/4}\approx 136.2332298227689$ | ||
Ramified primes: | \(3\), \(337\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{337}) \) | ||
$\card{ \Gal(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(1011=3\cdot 337\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1011}(1,·)$, $\chi_{1011}(485,·)$, $\chi_{1011}(673,·)$, $\chi_{1011}(863,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | 4.0.344454777.2$^{2}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{1544}a^{3}+\frac{585}{1544}a^{2}+\frac{253}{1544}a-\frac{239}{772}$
Monogenic: | No | |
Index: | $2$ | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{10}$, which has order $20$
Unit group
Rank: | $1$ | 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 unit: | $\frac{110671282}{193}a^{3}+\frac{664027692}{193}a^{2}+\frac{6640276920}{193}a-\frac{24680195671}{193}$ | 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: | \( 42.8642328147 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{2}\cdot 42.8642328147 \cdot 20}{2\cdot\sqrt{344454777}}\cr\approx \mathstrut & 0.911777111445 \end{aligned}\]
Galois group
A cyclic group of order 4 |
The 4 conjugacy class representatives for $C_4$ |
Character table for $C_4$ |
Intermediate fields
\(\Q(\sqrt{337}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.4.0.1}{4} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(337\) | Deg $4$ | $4$ | $1$ | $3$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.1011.4t1.a.a | $1$ | $ 3 \cdot 337 $ | 4.0.344454777.2 | $C_4$ (as 4T1) | $0$ | $-1$ |
* | 1.337.2t1.a.a | $1$ | $ 337 $ | \(\Q(\sqrt{337}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.1011.4t1.a.b | $1$ | $ 3 \cdot 337 $ | 4.0.344454777.2 | $C_4$ (as 4T1) | $0$ | $-1$ |