Normalized defining polynomial
\( x^{12} - 3 x^{11} + 21 x^{10} - 19 x^{9} + 180 x^{8} - 54 x^{7} + 1899 x^{6} - 72 x^{5} + 10584 x^{4} + 7056 x^{3} + 34830 x^{2} + 16929 x + 29241 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(5514297181968073041\) \(\medspace = 3^{16}\cdot 71^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(36.46\) | 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: | $3^{4/3}71^{1/2}\approx 36.45783266912841$ | ||
Ramified primes: | \(3\), \(71\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-71}) \), 6.0.2348254071.3$^{3}$, 8.0.166726039041.1$^{4}$, 12.0.5514297181968073041.1$^{12}$, deg 24$^{12}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}$, $\frac{1}{6}a^{6}-\frac{1}{6}a^{5}-\frac{1}{6}a^{4}+\frac{1}{6}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{6}a^{7}+\frac{1}{3}a^{4}-\frac{1}{2}$, $\frac{1}{6}a^{8}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{2}a$, $\frac{1}{36}a^{9}-\frac{1}{12}a^{8}-\frac{1}{36}a^{6}+\frac{1}{12}a^{5}+\frac{5}{12}a^{4}-\frac{5}{12}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}$, $\frac{1}{684}a^{10}+\frac{1}{76}a^{9}-\frac{7}{114}a^{8}+\frac{47}{684}a^{7}+\frac{1}{228}a^{6}+\frac{13}{228}a^{5}-\frac{3}{76}a^{4}-\frac{47}{114}a^{3}-\frac{9}{19}a^{2}-\frac{9}{76}a$, $\frac{1}{25\!\cdots\!32}a^{11}+\frac{79810969615}{285918425892948}a^{10}+\frac{128063097587}{47653070982158}a^{9}+\frac{161996428977911}{25\!\cdots\!32}a^{8}-\frac{41946560143423}{857755277678844}a^{7}-\frac{54387048443423}{857755277678844}a^{6}-\frac{6064726196945}{95306141964316}a^{5}-\frac{33872880848233}{142959212946474}a^{4}-\frac{10831891068571}{142959212946474}a^{3}+\frac{38875342146499}{285918425892948}a^{2}-\frac{3191464371175}{23826535491079}a+\frac{298563280010}{1254028183741}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $3$ |
Class group and class number
$C_{42}$, which has order $42$
Unit group
Rank: | $5$ | 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: | $\frac{5138780135}{135435043844028}a^{11}-\frac{14217269575}{45145014614676}a^{10}+\frac{63736923233}{45145014614676}a^{9}-\frac{348555118507}{67717521922014}a^{8}+\frac{514427985389}{45145014614676}a^{7}-\frac{96281226977}{2508056367482}a^{6}+\frac{306304999885}{3762084551223}a^{5}-\frac{5872851308009}{15048338204892}a^{4}+\frac{2409013160525}{5016112734964}a^{3}-\frac{27173317198147}{15048338204892}a^{2}-\frac{333193393543}{2508056367482}a-\frac{25949353248749}{5016112734964}$, $\frac{5138780135}{135435043844028}a^{11}-\frac{14217269575}{45145014614676}a^{10}+\frac{63736923233}{45145014614676}a^{9}-\frac{348555118507}{67717521922014}a^{8}+\frac{514427985389}{45145014614676}a^{7}-\frac{96281226977}{2508056367482}a^{6}+\frac{306304999885}{3762084551223}a^{5}-\frac{5872851308009}{15048338204892}a^{4}+\frac{2409013160525}{5016112734964}a^{3}-\frac{27173317198147}{15048338204892}a^{2}-\frac{333193393543}{2508056367482}a-\frac{20933240513785}{5016112734964}$, $\frac{39920012449}{643316458259133}a^{11}-\frac{248440842305}{857755277678844}a^{10}+\frac{356839133891}{428877638839422}a^{9}-\frac{4997342235421}{25\!\cdots\!32}a^{8}+\frac{3282158760901}{857755277678844}a^{7}-\frac{970495177375}{47653070982158}a^{6}+\frac{2448818714641}{47653070982158}a^{5}-\frac{3806218965536}{23826535491079}a^{4}-\frac{11150419342155}{95306141964316}a^{3}-\frac{121975325641303}{142959212946474}a^{2}-\frac{2393026863789}{5016112734964}a-\frac{4579783857011}{5016112734964}$, $\frac{76142989163}{857755277678844}a^{11}-\frac{142816154947}{214438819419711}a^{10}+\frac{930081425755}{285918425892948}a^{9}-\frac{9863237471867}{857755277678844}a^{8}+\frac{12502896769403}{428877638839422}a^{7}-\frac{11260744817497}{142959212946474}a^{6}+\frac{4591668456524}{23826535491079}a^{5}-\frac{222304740039869}{285918425892948}a^{4}+\frac{33137837663018}{23826535491079}a^{3}-\frac{342895169722267}{95306141964316}a^{2}+\frac{855295553997}{5016112734964}a-\frac{6226104984083}{1254028183741}$, $\frac{132281728943}{25\!\cdots\!32}a^{11}-\frac{26593306973}{142959212946474}a^{10}+\frac{349827219419}{857755277678844}a^{9}-\frac{1194007972541}{25\!\cdots\!32}a^{8}+\frac{300230051365}{214438819419711}a^{7}-\frac{400183131036}{23826535491079}a^{6}+\frac{6461692410793}{142959212946474}a^{5}-\frac{43312620965857}{285918425892948}a^{4}-\frac{4028799952867}{23826535491079}a^{3}-\frac{242814895562665}{285918425892948}a^{2}-\frac{2675684129133}{5016112734964}a-\frac{565957455733}{1254028183741}$ | 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: | \( 3039.60843665 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 3039.60843665 \cdot 42}{2\cdot\sqrt{5514297181968073041}}\cr\approx \mathstrut & 1.67251900817 \end{aligned}\]
Galois group
$C_2\times A_4$ (as 12T7):
A solvable group of order 24 |
The 8 conjugacy class representatives for $A_4 \times C_2$ |
Character table for $A_4 \times C_2$ |
Intermediate fields
\(\Q(\sqrt{-71}) \), \(\Q(\zeta_{9})^+\), 6.0.2348254071.3, 6.0.465831.1, 6.6.33074001.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 6 sibling: | 6.0.465831.1 |
Degree 8 sibling: | 8.0.166726039041.1 |
Degree 12 sibling: | 12.0.1093889542148001.2 |
Minimal sibling: | 6.0.465831.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
\(71\) | 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.4.2.1 | $x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
71.4.2.1 | $x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |