Properties

Label 46.46.962...941.1
Degree $46$
Signature $[46, 0]$
Discriminant $9.624\times 10^{103}$
Root discriminant \(182.18\)
Ramified primes $3,7,47$
Class number not computed
Class group not computed
Galois group $C_{46}$ (as 46T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^46 - 21*x^45 + 50*x^44 + 1883*x^43 - 12920*x^42 - 60765*x^41 + 785503*x^40 + 300187*x^39 - 25575587*x^38 + 40214110*x^37 + 521735008*x^36 - 1564306550*x^35 - 6949929857*x^34 + 31787689566*x^33 + 57756246374*x^32 - 425729426952*x^31 - 210107630718*x^30 + 4019949670704*x^29 - 1386075895421*x^28 - 27371507035122*x^27 + 26650221853526*x^26 + 134194010274701*x^25 - 206486899342915*x^24 - 461722534108312*x^23 + 1009252077539639*x^22 + 1036539452322248*x^21 - 3366962181446806*x^20 - 1164849476322639*x^19 + 7767671494048670*x^18 - 741931405771523*x^17 - 12201690528858935*x^16 + 5045104955656677*x^15 + 12550562657707220*x^14 - 8456879148847524*x^13 - 7865485324249473*x^12 + 7415009326963335*x^11 + 2594178944982243*x^10 - 3580244990409732*x^9 - 271735536883800*x^8 + 897997567222545*x^7 - 44388322932266*x^6 - 100983278491909*x^5 + 8398133311471*x^4 + 3441355105333*x^3 - 332318327837*x^2 - 17482059720*x + 1254092071)
 
gp: K = bnfinit(y^46 - 21*y^45 + 50*y^44 + 1883*y^43 - 12920*y^42 - 60765*y^41 + 785503*y^40 + 300187*y^39 - 25575587*y^38 + 40214110*y^37 + 521735008*y^36 - 1564306550*y^35 - 6949929857*y^34 + 31787689566*y^33 + 57756246374*y^32 - 425729426952*y^31 - 210107630718*y^30 + 4019949670704*y^29 - 1386075895421*y^28 - 27371507035122*y^27 + 26650221853526*y^26 + 134194010274701*y^25 - 206486899342915*y^24 - 461722534108312*y^23 + 1009252077539639*y^22 + 1036539452322248*y^21 - 3366962181446806*y^20 - 1164849476322639*y^19 + 7767671494048670*y^18 - 741931405771523*y^17 - 12201690528858935*y^16 + 5045104955656677*y^15 + 12550562657707220*y^14 - 8456879148847524*y^13 - 7865485324249473*y^12 + 7415009326963335*y^11 + 2594178944982243*y^10 - 3580244990409732*y^9 - 271735536883800*y^8 + 897997567222545*y^7 - 44388322932266*y^6 - 100983278491909*y^5 + 8398133311471*y^4 + 3441355105333*y^3 - 332318327837*y^2 - 17482059720*y + 1254092071, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^46 - 21*x^45 + 50*x^44 + 1883*x^43 - 12920*x^42 - 60765*x^41 + 785503*x^40 + 300187*x^39 - 25575587*x^38 + 40214110*x^37 + 521735008*x^36 - 1564306550*x^35 - 6949929857*x^34 + 31787689566*x^33 + 57756246374*x^32 - 425729426952*x^31 - 210107630718*x^30 + 4019949670704*x^29 - 1386075895421*x^28 - 27371507035122*x^27 + 26650221853526*x^26 + 134194010274701*x^25 - 206486899342915*x^24 - 461722534108312*x^23 + 1009252077539639*x^22 + 1036539452322248*x^21 - 3366962181446806*x^20 - 1164849476322639*x^19 + 7767671494048670*x^18 - 741931405771523*x^17 - 12201690528858935*x^16 + 5045104955656677*x^15 + 12550562657707220*x^14 - 8456879148847524*x^13 - 7865485324249473*x^12 + 7415009326963335*x^11 + 2594178944982243*x^10 - 3580244990409732*x^9 - 271735536883800*x^8 + 897997567222545*x^7 - 44388322932266*x^6 - 100983278491909*x^5 + 8398133311471*x^4 + 3441355105333*x^3 - 332318327837*x^2 - 17482059720*x + 1254092071);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^46 - 21*x^45 + 50*x^44 + 1883*x^43 - 12920*x^42 - 60765*x^41 + 785503*x^40 + 300187*x^39 - 25575587*x^38 + 40214110*x^37 + 521735008*x^36 - 1564306550*x^35 - 6949929857*x^34 + 31787689566*x^33 + 57756246374*x^32 - 425729426952*x^31 - 210107630718*x^30 + 4019949670704*x^29 - 1386075895421*x^28 - 27371507035122*x^27 + 26650221853526*x^26 + 134194010274701*x^25 - 206486899342915*x^24 - 461722534108312*x^23 + 1009252077539639*x^22 + 1036539452322248*x^21 - 3366962181446806*x^20 - 1164849476322639*x^19 + 7767671494048670*x^18 - 741931405771523*x^17 - 12201690528858935*x^16 + 5045104955656677*x^15 + 12550562657707220*x^14 - 8456879148847524*x^13 - 7865485324249473*x^12 + 7415009326963335*x^11 + 2594178944982243*x^10 - 3580244990409732*x^9 - 271735536883800*x^8 + 897997567222545*x^7 - 44388322932266*x^6 - 100983278491909*x^5 + 8398133311471*x^4 + 3441355105333*x^3 - 332318327837*x^2 - 17482059720*x + 1254092071)
 

\( x^{46} - 21 x^{45} + 50 x^{44} + 1883 x^{43} - 12920 x^{42} - 60765 x^{41} + 785503 x^{40} + 300187 x^{39} - 25575587 x^{38} + 40214110 x^{37} + 521735008 x^{36} + \cdots + 1254092071 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $46$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[46, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(962\!\cdots\!941\) \(\medspace = 3^{23}\cdot 7^{23}\cdot 47^{44}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(182.18\)
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:  $3^{1/2}7^{1/2}47^{22/23}\approx 182.18289476147334$
Ramified primes:   \(3\), \(7\), \(47\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{21}) \)
$\card{ \Gal(K/\Q) }$:  $46$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(987=3\cdot 7\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{987}(1,·)$, $\chi_{987}(902,·)$, $\chi_{987}(776,·)$, $\chi_{987}(209,·)$, $\chi_{987}(524,·)$, $\chi_{987}(526,·)$, $\chi_{987}(272,·)$, $\chi_{987}(148,·)$, $\chi_{987}(925,·)$, $\chi_{987}(545,·)$, $\chi_{987}(169,·)$, $\chi_{987}(944,·)$, $\chi_{987}(946,·)$, $\chi_{987}(883,·)$, $\chi_{987}(820,·)$, $\chi_{987}(566,·)$, $\chi_{987}(841,·)$, $\chi_{987}(440,·)$, $\chi_{987}(692,·)$, $\chi_{987}(314,·)$, $\chi_{987}(316,·)$, $\chi_{987}(190,·)$, $\chi_{987}(64,·)$, $\chi_{987}(965,·)$, $\chi_{987}(967,·)$, $\chi_{987}(713,·)$, $\chi_{987}(589,·)$, $\chi_{987}(335,·)$, $\chi_{987}(568,·)$, $\chi_{987}(83,·)$, $\chi_{987}(356,·)$, $\chi_{987}(860,·)$, $\chi_{987}(862,·)$, $\chi_{987}(400,·)$, $\chi_{987}(482,·)$, $\chi_{987}(379,·)$, $\chi_{987}(484,·)$, $\chi_{987}(230,·)$, $\chi_{987}(337,·)$, $\chi_{987}(106,·)$, $\chi_{987}(694,·)$, $\chi_{987}(755,·)$, $\chi_{987}(629,·)$, $\chi_{987}(377,·)$, $\chi_{987}(251,·)$, $\chi_{987}(253,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $a^{44}$, $\frac{1}{47\!\cdots\!09}a^{45}-\frac{28\!\cdots\!47}{47\!\cdots\!09}a^{44}+\frac{10\!\cdots\!16}{47\!\cdots\!09}a^{43}-\frac{66\!\cdots\!25}{47\!\cdots\!09}a^{42}-\frac{31\!\cdots\!86}{47\!\cdots\!09}a^{41}-\frac{95\!\cdots\!56}{47\!\cdots\!09}a^{40}-\frac{87\!\cdots\!27}{47\!\cdots\!09}a^{39}+\frac{99\!\cdots\!43}{47\!\cdots\!09}a^{38}-\frac{23\!\cdots\!56}{47\!\cdots\!09}a^{37}+\frac{55\!\cdots\!21}{47\!\cdots\!09}a^{36}+\frac{13\!\cdots\!72}{47\!\cdots\!09}a^{35}-\frac{12\!\cdots\!86}{47\!\cdots\!09}a^{34}-\frac{87\!\cdots\!26}{47\!\cdots\!09}a^{33}+\frac{17\!\cdots\!00}{47\!\cdots\!09}a^{32}-\frac{35\!\cdots\!17}{47\!\cdots\!09}a^{31}+\frac{55\!\cdots\!99}{47\!\cdots\!09}a^{30}-\frac{16\!\cdots\!18}{47\!\cdots\!09}a^{29}-\frac{20\!\cdots\!77}{47\!\cdots\!09}a^{28}+\frac{26\!\cdots\!63}{47\!\cdots\!09}a^{27}+\frac{50\!\cdots\!14}{47\!\cdots\!09}a^{26}+\frac{14\!\cdots\!19}{47\!\cdots\!09}a^{25}+\frac{61\!\cdots\!87}{47\!\cdots\!09}a^{24}+\frac{51\!\cdots\!79}{47\!\cdots\!09}a^{23}+\frac{22\!\cdots\!06}{47\!\cdots\!09}a^{22}+\frac{78\!\cdots\!35}{47\!\cdots\!09}a^{21}+\frac{26\!\cdots\!23}{47\!\cdots\!09}a^{20}+\frac{20\!\cdots\!36}{47\!\cdots\!09}a^{19}-\frac{81\!\cdots\!57}{47\!\cdots\!09}a^{18}+\frac{13\!\cdots\!03}{47\!\cdots\!09}a^{17}-\frac{18\!\cdots\!22}{47\!\cdots\!09}a^{16}+\frac{22\!\cdots\!60}{47\!\cdots\!09}a^{15}+\frac{16\!\cdots\!00}{47\!\cdots\!09}a^{14}+\frac{19\!\cdots\!41}{47\!\cdots\!09}a^{13}+\frac{44\!\cdots\!71}{47\!\cdots\!09}a^{12}-\frac{51\!\cdots\!18}{47\!\cdots\!09}a^{11}+\frac{86\!\cdots\!65}{47\!\cdots\!09}a^{10}-\frac{28\!\cdots\!41}{47\!\cdots\!09}a^{9}-\frac{21\!\cdots\!51}{47\!\cdots\!09}a^{8}-\frac{21\!\cdots\!96}{47\!\cdots\!09}a^{7}-\frac{15\!\cdots\!38}{47\!\cdots\!09}a^{6}-\frac{17\!\cdots\!79}{47\!\cdots\!09}a^{5}+\frac{52\!\cdots\!17}{47\!\cdots\!09}a^{4}-\frac{58\!\cdots\!72}{47\!\cdots\!09}a^{3}+\frac{67\!\cdots\!11}{47\!\cdots\!09}a^{2}+\frac{18\!\cdots\!85}{47\!\cdots\!09}a+\frac{20\!\cdots\!92}{47\!\cdots\!09}$ Copy content Toggle raw display

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

not computed

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:  $45$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^46 - 21*x^45 + 50*x^44 + 1883*x^43 - 12920*x^42 - 60765*x^41 + 785503*x^40 + 300187*x^39 - 25575587*x^38 + 40214110*x^37 + 521735008*x^36 - 1564306550*x^35 - 6949929857*x^34 + 31787689566*x^33 + 57756246374*x^32 - 425729426952*x^31 - 210107630718*x^30 + 4019949670704*x^29 - 1386075895421*x^28 - 27371507035122*x^27 + 26650221853526*x^26 + 134194010274701*x^25 - 206486899342915*x^24 - 461722534108312*x^23 + 1009252077539639*x^22 + 1036539452322248*x^21 - 3366962181446806*x^20 - 1164849476322639*x^19 + 7767671494048670*x^18 - 741931405771523*x^17 - 12201690528858935*x^16 + 5045104955656677*x^15 + 12550562657707220*x^14 - 8456879148847524*x^13 - 7865485324249473*x^12 + 7415009326963335*x^11 + 2594178944982243*x^10 - 3580244990409732*x^9 - 271735536883800*x^8 + 897997567222545*x^7 - 44388322932266*x^6 - 100983278491909*x^5 + 8398133311471*x^4 + 3441355105333*x^3 - 332318327837*x^2 - 17482059720*x + 1254092071)
 
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^46 - 21*x^45 + 50*x^44 + 1883*x^43 - 12920*x^42 - 60765*x^41 + 785503*x^40 + 300187*x^39 - 25575587*x^38 + 40214110*x^37 + 521735008*x^36 - 1564306550*x^35 - 6949929857*x^34 + 31787689566*x^33 + 57756246374*x^32 - 425729426952*x^31 - 210107630718*x^30 + 4019949670704*x^29 - 1386075895421*x^28 - 27371507035122*x^27 + 26650221853526*x^26 + 134194010274701*x^25 - 206486899342915*x^24 - 461722534108312*x^23 + 1009252077539639*x^22 + 1036539452322248*x^21 - 3366962181446806*x^20 - 1164849476322639*x^19 + 7767671494048670*x^18 - 741931405771523*x^17 - 12201690528858935*x^16 + 5045104955656677*x^15 + 12550562657707220*x^14 - 8456879148847524*x^13 - 7865485324249473*x^12 + 7415009326963335*x^11 + 2594178944982243*x^10 - 3580244990409732*x^9 - 271735536883800*x^8 + 897997567222545*x^7 - 44388322932266*x^6 - 100983278491909*x^5 + 8398133311471*x^4 + 3441355105333*x^3 - 332318327837*x^2 - 17482059720*x + 1254092071, 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^46 - 21*x^45 + 50*x^44 + 1883*x^43 - 12920*x^42 - 60765*x^41 + 785503*x^40 + 300187*x^39 - 25575587*x^38 + 40214110*x^37 + 521735008*x^36 - 1564306550*x^35 - 6949929857*x^34 + 31787689566*x^33 + 57756246374*x^32 - 425729426952*x^31 - 210107630718*x^30 + 4019949670704*x^29 - 1386075895421*x^28 - 27371507035122*x^27 + 26650221853526*x^26 + 134194010274701*x^25 - 206486899342915*x^24 - 461722534108312*x^23 + 1009252077539639*x^22 + 1036539452322248*x^21 - 3366962181446806*x^20 - 1164849476322639*x^19 + 7767671494048670*x^18 - 741931405771523*x^17 - 12201690528858935*x^16 + 5045104955656677*x^15 + 12550562657707220*x^14 - 8456879148847524*x^13 - 7865485324249473*x^12 + 7415009326963335*x^11 + 2594178944982243*x^10 - 3580244990409732*x^9 - 271735536883800*x^8 + 897997567222545*x^7 - 44388322932266*x^6 - 100983278491909*x^5 + 8398133311471*x^4 + 3441355105333*x^3 - 332318327837*x^2 - 17482059720*x + 1254092071);
 
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^46 - 21*x^45 + 50*x^44 + 1883*x^43 - 12920*x^42 - 60765*x^41 + 785503*x^40 + 300187*x^39 - 25575587*x^38 + 40214110*x^37 + 521735008*x^36 - 1564306550*x^35 - 6949929857*x^34 + 31787689566*x^33 + 57756246374*x^32 - 425729426952*x^31 - 210107630718*x^30 + 4019949670704*x^29 - 1386075895421*x^28 - 27371507035122*x^27 + 26650221853526*x^26 + 134194010274701*x^25 - 206486899342915*x^24 - 461722534108312*x^23 + 1009252077539639*x^22 + 1036539452322248*x^21 - 3366962181446806*x^20 - 1164849476322639*x^19 + 7767671494048670*x^18 - 741931405771523*x^17 - 12201690528858935*x^16 + 5045104955656677*x^15 + 12550562657707220*x^14 - 8456879148847524*x^13 - 7865485324249473*x^12 + 7415009326963335*x^11 + 2594178944982243*x^10 - 3580244990409732*x^9 - 271735536883800*x^8 + 897997567222545*x^7 - 44388322932266*x^6 - 100983278491909*x^5 + 8398133311471*x^4 + 3441355105333*x^3 - 332318327837*x^2 - 17482059720*x + 1254092071);
 
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

$C_{46}$ (as 46T1):

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 cyclic group of order 46
The 46 conjugacy class representatives for $C_{46}$
Character table for $C_{46}$ is not computed

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\zeta_{47})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

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 $46$ R $23^{2}$ R $46$ $46$ $23^{2}$ $46$ $46$ $46$ $46$ $23^{2}$ $23^{2}$ $23^{2}$ R $46$ $23^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display Deg $46$$2$$23$$23$
\(7\) Copy content Toggle raw display Deg $46$$2$$23$$23$
\(47\) Copy content Toggle raw display 47.23.22.1$x^{23} + 47$$23$$1$$22$$C_{23}$$[\ ]_{23}$
47.23.22.1$x^{23} + 47$$23$$1$$22$$C_{23}$$[\ ]_{23}$