Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^31 + 4*x - 3)
gp: K = bnfinit(y^31 + 4*y - 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^31 + 4*x - 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^31 + 4*x - 3)
\( x^{31} + 4x - 3 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $31$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-19449380546628051604251457953600887142088240597897609060431831\)
\(\medspace = -\,3^{30}\cdot 59\cdot 10709\cdot 14\!\cdots\!49\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(94.86\) | 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: | not computed | ||
| Ramified primes: |
\(3\), \(59\), \(10709\), \(14950\!\cdots\!07849\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-94464\!\cdots\!41519}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{7}a^{30}+\frac{2}{7}a^{29}-\frac{3}{7}a^{28}+\frac{1}{7}a^{27}+\frac{2}{7}a^{26}-\frac{3}{7}a^{25}+\frac{1}{7}a^{24}+\frac{2}{7}a^{23}-\frac{3}{7}a^{22}+\frac{1}{7}a^{21}+\frac{2}{7}a^{20}-\frac{3}{7}a^{19}+\frac{1}{7}a^{18}+\frac{2}{7}a^{17}-\frac{3}{7}a^{16}+\frac{1}{7}a^{15}+\frac{2}{7}a^{14}-\frac{3}{7}a^{13}+\frac{1}{7}a^{12}+\frac{2}{7}a^{11}-\frac{3}{7}a^{10}+\frac{1}{7}a^{9}+\frac{2}{7}a^{8}-\frac{3}{7}a^{7}+\frac{1}{7}a^{6}+\frac{2}{7}a^{5}-\frac{3}{7}a^{4}+\frac{1}{7}a^{3}+\frac{2}{7}a^{2}-\frac{3}{7}a-\frac{2}{7}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $15$ | 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: |
$a^{20}-a^{19}+a^{18}-a^{17}+a^{16}-a^{15}+a^{14}-a^{13}+a^{12}-a^{11}+2a^{10}-2a^{9}+2a^{8}-2a^{7}+2a^{6}-2a^{5}+2a^{4}-2a^{3}+2a^{2}-2a+1$, $a^{28}-a^{27}+2a^{26}-2a^{25}+2a^{24}-2a^{23}+a^{22}-a^{21}+a^{16}-a^{15}+2a^{14}-2a^{13}+2a^{12}-2a^{11}+a^{10}-a^{9}+a^{4}-a^{3}+2a^{2}-2a+1$, $2a^{22}-a^{21}-3a^{12}+2a^{11}+5a^{2}-5a+1$, $\frac{12}{7}a^{30}+\frac{10}{7}a^{29}-\frac{1}{7}a^{28}+\frac{5}{7}a^{27}+\frac{3}{7}a^{26}-\frac{1}{7}a^{25}+\frac{12}{7}a^{24}+\frac{3}{7}a^{23}-\frac{8}{7}a^{22}+\frac{5}{7}a^{21}-\frac{4}{7}a^{20}-\frac{1}{7}a^{19}+\frac{12}{7}a^{18}-\frac{11}{7}a^{17}-\frac{15}{7}a^{16}+\frac{12}{7}a^{15}-\frac{18}{7}a^{14}+\frac{6}{7}a^{13}+\frac{12}{7}a^{12}-\frac{46}{7}a^{11}+\frac{13}{7}a^{10}+\frac{5}{7}a^{9}-\frac{32}{7}a^{8}+\frac{27}{7}a^{7}-\frac{16}{7}a^{6}-\frac{46}{7}a^{5}+\frac{41}{7}a^{4}-\frac{23}{7}a^{3}-\frac{18}{7}a^{2}+\frac{41}{7}a-\frac{10}{7}$, $\frac{79}{7}a^{30}+\frac{74}{7}a^{29}+\frac{1}{7}a^{28}-\frac{82}{7}a^{27}-\frac{94}{7}a^{26}-\frac{20}{7}a^{25}+\frac{79}{7}a^{24}+\frac{116}{7}a^{23}+\frac{57}{7}a^{22}-\frac{61}{7}a^{21}-\frac{129}{7}a^{20}-\frac{83}{7}a^{19}+\frac{44}{7}a^{18}+\frac{144}{7}a^{17}+\frac{127}{7}a^{16}-\frac{5}{7}a^{15}-\frac{143}{7}a^{14}-\frac{160}{7}a^{13}-\frac{40}{7}a^{12}+\frac{130}{7}a^{11}+\frac{197}{7}a^{10}+\frac{86}{7}a^{9}-\frac{115}{7}a^{8}-\frac{223}{7}a^{7}-\frac{145}{7}a^{6}+\frac{60}{7}a^{5}+\frac{239}{7}a^{4}+\frac{205}{7}a^{3}-\frac{24}{7}a^{2}-\frac{244}{7}a+\frac{59}{7}$, $\frac{129}{7}a^{30}+\frac{111}{7}a^{29}+\frac{103}{7}a^{28}+\frac{80}{7}a^{27}+\frac{48}{7}a^{26}+\frac{40}{7}a^{25}+\frac{52}{7}a^{24}+\frac{62}{7}a^{23}+\frac{26}{7}a^{22}+\frac{3}{7}a^{21}+\frac{27}{7}a^{20}+\frac{54}{7}a^{19}+\frac{24}{7}a^{18}-\frac{8}{7}a^{17}+\frac{5}{7}a^{16}+\frac{24}{7}a^{15}+\frac{34}{7}a^{14}-\frac{2}{7}a^{13}-\frac{25}{7}a^{12}-\frac{15}{7}a^{11}+\frac{26}{7}a^{10}+\frac{17}{7}a^{9}-\frac{43}{7}a^{8}-\frac{65}{7}a^{7}+\frac{3}{7}a^{6}+\frac{41}{7}a^{5}-\frac{51}{7}a^{4}-\frac{81}{7}a^{3}-\frac{29}{7}a^{2}+\frac{26}{7}a+\frac{505}{7}$, $\frac{15}{7}a^{30}-\frac{12}{7}a^{29}+\frac{4}{7}a^{28}+\frac{22}{7}a^{27}-\frac{12}{7}a^{26}-\frac{31}{7}a^{25}+\frac{15}{7}a^{24}+\frac{51}{7}a^{23}-\frac{3}{7}a^{22}-\frac{55}{7}a^{21}-\frac{12}{7}a^{20}+\frac{32}{7}a^{19}+\frac{1}{7}a^{18}-\frac{26}{7}a^{17}+\frac{32}{7}a^{16}+\frac{50}{7}a^{15}-\frac{40}{7}a^{14}-\frac{80}{7}a^{13}+\frac{22}{7}a^{12}+\frac{86}{7}a^{11}-\frac{3}{7}a^{10}-\frac{69}{7}a^{9}+\frac{2}{7}a^{8}+\frac{39}{7}a^{7}-\frac{62}{7}a^{6}-\frac{61}{7}a^{5}+\frac{95}{7}a^{4}+\frac{127}{7}a^{3}-\frac{61}{7}a^{2}-\frac{143}{7}a+\frac{82}{7}$, $\frac{138}{7}a^{30}+\frac{45}{7}a^{29}+\frac{83}{7}a^{28}+\frac{68}{7}a^{27}-\frac{18}{7}a^{26}+\frac{69}{7}a^{25}+\frac{33}{7}a^{24}-\frac{46}{7}a^{23}+\frac{69}{7}a^{22}+\frac{5}{7}a^{21}-\frac{46}{7}a^{20}+\frac{76}{7}a^{19}-\frac{37}{7}a^{18}-\frac{25}{7}a^{17}+\frac{90}{7}a^{16}-\frac{86}{7}a^{15}+\frac{3}{7}a^{14}+\frac{97}{7}a^{13}-\frac{121}{7}a^{12}+\frac{38}{7}a^{11}+\frac{83}{7}a^{10}-\frac{142}{7}a^{9}+\frac{94}{7}a^{8}+\frac{41}{7}a^{7}-\frac{156}{7}a^{6}+\frac{157}{7}a^{5}-\frac{15}{7}a^{4}-\frac{149}{7}a^{3}+\frac{199}{7}a^{2}-\frac{85}{7}a+\frac{445}{7}$, $3a^{30}-3a^{28}-5a^{27}+a^{26}+6a^{25}+4a^{24}+2a^{23}-2a^{22}-8a^{21}-3a^{20}+5a^{19}+5a^{18}+5a^{17}-9a^{15}-7a^{14}+3a^{13}+7a^{12}+8a^{11}+4a^{10}-7a^{9}-12a^{8}-4a^{7}+7a^{6}+9a^{5}+9a^{4}-16a^{2}-12a+19$, $\frac{68}{7}a^{30}-\frac{102}{7}a^{29}-\frac{183}{7}a^{28}-\frac{100}{7}a^{27}+\frac{87}{7}a^{26}+\frac{223}{7}a^{25}+\frac{173}{7}a^{24}-\frac{32}{7}a^{23}-\frac{211}{7}a^{22}-\frac{212}{7}a^{21}-\frac{18}{7}a^{20}+\frac{216}{7}a^{19}+\frac{292}{7}a^{18}+\frac{108}{7}a^{17}-\frac{183}{7}a^{16}-\frac{324}{7}a^{15}-\frac{179}{7}a^{14}+\frac{146}{7}a^{13}+\frac{369}{7}a^{12}+\frac{297}{7}a^{11}-\frac{64}{7}a^{10}-\frac{394}{7}a^{9}-\frac{382}{7}a^{8}-\frac{15}{7}a^{7}+\frac{383}{7}a^{6}+\frac{472}{7}a^{5}+\frac{153}{7}a^{4}-\frac{352}{7}a^{3}-\frac{571}{7}a^{2}-\frac{288}{7}a+\frac{550}{7}$, $\frac{39}{7}a^{30}-\frac{13}{7}a^{29}+\frac{37}{7}a^{28}-\frac{3}{7}a^{27}+\frac{15}{7}a^{26}+\frac{23}{7}a^{25}+\frac{4}{7}a^{24}+\frac{29}{7}a^{23}+\frac{9}{7}a^{22}+\frac{11}{7}a^{21}+\frac{1}{7}a^{20}+\frac{30}{7}a^{19}-\frac{24}{7}a^{18}+\frac{64}{7}a^{17}-\frac{47}{7}a^{16}+\frac{46}{7}a^{15}-\frac{13}{7}a^{14}-\frac{26}{7}a^{13}+\frac{53}{7}a^{12}-\frac{62}{7}a^{11}+\frac{65}{7}a^{10}-\frac{52}{7}a^{9}+\frac{15}{7}a^{8}-\frac{33}{7}a^{7}+\frac{18}{7}a^{6}-\frac{55}{7}a^{5}+\frac{51}{7}a^{4}-\frac{52}{7}a^{3}-\frac{13}{7}a^{2}+\frac{37}{7}a-\frac{1}{7}$, $\frac{155}{7}a^{30}+\frac{142}{7}a^{29}+\frac{137}{7}a^{28}+\frac{127}{7}a^{27}+\frac{114}{7}a^{26}+\frac{109}{7}a^{25}+\frac{78}{7}a^{24}+\frac{72}{7}a^{23}+\frac{67}{7}a^{22}+\frac{50}{7}a^{21}+\frac{37}{7}a^{20}-\frac{3}{7}a^{19}-\frac{20}{7}a^{18}-\frac{26}{7}a^{17}-\frac{59}{7}a^{16}-\frac{69}{7}a^{15}-\frac{82}{7}a^{14}-\frac{94}{7}a^{13}-\frac{76}{7}a^{12}-\frac{124}{7}a^{11}-\frac{143}{7}a^{10}-\frac{153}{7}a^{9}-\frac{173}{7}a^{8}-\frac{143}{7}a^{7}-\frac{181}{7}a^{6}-\frac{173}{7}a^{5}-\frac{136}{7}a^{4}-\frac{146}{7}a^{3}-\frac{138}{7}a^{2}-\frac{164}{7}a+\frac{446}{7}$, $\frac{19}{7}a^{30}+\frac{24}{7}a^{29}+\frac{6}{7}a^{28}-\frac{2}{7}a^{27}+\frac{10}{7}a^{26}+\frac{13}{7}a^{25}+\frac{5}{7}a^{24}+\frac{3}{7}a^{23}-\frac{1}{7}a^{22}-\frac{2}{7}a^{21}+\frac{10}{7}a^{20}-\frac{8}{7}a^{19}-\frac{16}{7}a^{18}+\frac{10}{7}a^{17}+\frac{20}{7}a^{16}-\frac{9}{7}a^{15}-\frac{4}{7}a^{14}+\frac{13}{7}a^{13}-\frac{2}{7}a^{12}+\frac{3}{7}a^{11}-\frac{1}{7}a^{10}-\frac{16}{7}a^{9}-\frac{4}{7}a^{8}+\frac{27}{7}a^{7}-\frac{9}{7}a^{6}-\frac{18}{7}a^{5}+\frac{6}{7}a^{4}-\frac{9}{7}a^{3}+\frac{3}{7}a^{2}+\frac{27}{7}a+\frac{67}{7}$, $\frac{122}{7}a^{30}-\frac{29}{7}a^{29}-\frac{79}{7}a^{28}+\frac{129}{7}a^{27}-\frac{162}{7}a^{26}+\frac{96}{7}a^{25}+\frac{59}{7}a^{24}-\frac{155}{7}a^{23}+\frac{208}{7}a^{22}-\frac{165}{7}a^{21}-\frac{8}{7}a^{20}+\frac{159}{7}a^{19}-\frac{263}{7}a^{18}+\frac{265}{7}a^{17}-\frac{65}{7}a^{16}-\frac{151}{7}a^{15}+\frac{293}{7}a^{14}-\frac{373}{7}a^{13}+\frac{206}{7}a^{12}+\frac{118}{7}a^{11}-\frac{338}{7}a^{10}+\frac{465}{7}a^{9}-\frac{379}{7}a^{8}+\frac{5}{7}a^{7}+\frac{360}{7}a^{6}-\frac{582}{7}a^{5}+\frac{579}{7}a^{4}-\frac{186}{7}a^{3}-\frac{323}{7}a^{2}+\frac{670}{7}a-\frac{335}{7}$, $\frac{51}{7}a^{30}+\frac{4}{7}a^{29}+\frac{64}{7}a^{28}-\frac{40}{7}a^{27}+\frac{11}{7}a^{26}-\frac{48}{7}a^{25}+\frac{72}{7}a^{24}-\frac{10}{7}a^{23}+\frac{50}{7}a^{22}-\frac{82}{7}a^{21}+\frac{18}{7}a^{20}-\frac{27}{7}a^{19}+\frac{79}{7}a^{18}-\frac{10}{7}a^{17}-\frac{6}{7}a^{16}-\frac{75}{7}a^{15}+\frac{32}{7}a^{14}+\frac{29}{7}a^{13}+\frac{44}{7}a^{12}-\frac{3}{7}a^{11}-\frac{104}{7}a^{10}+\frac{16}{7}a^{9}+\frac{4}{7}a^{8}+\frac{127}{7}a^{7}-\frac{75}{7}a^{6}+\frac{32}{7}a^{5}-\frac{188}{7}a^{4}+\frac{170}{7}a^{3}-\frac{66}{7}a^{2}+\frac{218}{7}a-\frac{74}{7}$
| 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: | \( 4933387750736738000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{15}\cdot 4933387750736738000 \cdot 1}{2\cdot\sqrt{19449380546628051604251457953600887142088240597897609060431831}}\cr\approx \mathstrut & 1.05048629209093 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^31 + 4*x - 3)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^31 + 4*x - 3, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^31 + 4*x - 3);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^31 + 4*x - 3);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A non-solvable group of order 8222838654177922817725562880000000 |
| The 6842 conjugacy class representatives for $S_{31}$ are not computed |
| Character table for $S_{31}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{6}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | $31$ | $17{,}\,{\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $31$ | $22{,}\,{\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $31$ | ${\href{/padicField/31.10.0.1}{10} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | $31$ | ${\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $26{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | R |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.6.6.1 | $x^{6} - 6 x^{5} + 24 x^{4} + 6 x^{3} + 18 x + 9$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ | |
| 3.12.12.18 | $x^{12} - 36 x^{11} + 438 x^{10} + 120 x^{9} - 4563 x^{8} + 1188 x^{7} + 22410 x^{6} + 10692 x^{5} - 17658 x^{4} - 3780 x^{3} + 6804 x^{2} - 1296 x + 81$ | $3$ | $4$ | $12$ | 12T46 | $[3/2, 3/2]_{2}^{4}$ | |
| 3.12.12.10 | $x^{12} - 24 x^{11} + 306 x^{10} - 2004 x^{9} + 7236 x^{8} - 4374 x^{7} - 1458 x^{6} + 5832 x^{5} - 1836 x^{3} + 324 x^{2} + 486 x + 81$ | $3$ | $4$ | $12$ | 12T173 | $[3/2, 3/2, 3/2, 3/2]_{2}^{4}$ | |
|
\(59\)
| 59.2.1.1 | $x^{2} + 118$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 59.4.0.1 | $x^{4} + 2 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 59.8.0.1 | $x^{8} + 16 x^{4} + 32 x^{3} + 2 x^{2} + 50 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 59.17.0.1 | $x^{17} + 9 x + 57$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
|
\(10709\)
| $\Q_{10709}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{10709}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{10709}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
|
\(149\!\cdots\!849\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ |