Properties

Label 4650bi3
Conductor 46504650
Discriminant 1.030×10131.030\times 10^{13}
j-invariant 32208729120020809658986840 \frac{32208729120020809}{658986840}
CM no
Rank 11
Torsion structure Z/2Z\Z/{2}\Z

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2+xy=x3165713x+25950417y^2+xy=x^3-165713x+25950417 Copy content Toggle raw display (homogenize, simplify)
y2z+xyz=x3165713xz2+25950417z3y^2z+xyz=x^3-165713xz^2+25950417z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x3214764075x+1211386947750y^2=x^3-214764075x+1211386947750 Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, 0, 0, -165713, 25950417])
 
gp: E = ellinit([1, 0, 0, -165713, 25950417])
 
magma: E := EllipticCurve([1, 0, 0, -165713, 25950417]);
 
oscar: E = elliptic_curve([1, 0, 0, -165713, 25950417])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

ZZ/2Z\Z \oplus \Z/{2}\Z

magma: MordellWeilGroup(E);
 

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(232,41)(232, -41)0.254549582511335356870888856200.25454958251133535687088885620\infty
(943/4,943/8)(943/4, -943/8)0022

Integral points

(68,6109) \left(-68, 6109\right) , (68,6041) \left(-68, -6041\right) , (232,41) \left(232, -41\right) , (232,191) \left(232, -191\right) , (256,439) \left(256, 439\right) , (256,695) \left(256, -695\right) , (292,1429) \left(292, 1429\right) , (292,1721) \left(292, -1721\right) , (2992,160729) \left(2992, 160729\right) , (2992,163721) \left(2992, -163721\right) Copy content Toggle raw display

comment: Integral points
 
sage: E.integral_points()
 
magma: IntegralPoints(E);
 

Invariants

Conductor: NN  =  4650 4650  = 2352312 \cdot 3 \cdot 5^{2} \cdot 31
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: Δ\Delta  =  1029666937500010296669375000 = 2331257312^{3} \cdot 3^{12} \cdot 5^{7} \cdot 31
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: jj  =  32208729120020809658986840 \frac{32208729120020809}{658986840}  = 2331251311373385332^{-3} \cdot 3^{-12} \cdot 5^{-1} \cdot 31^{-1} \cdot 373^{3} \cdot 853^{3}
comment: j-invariant
 
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
oscar: j_invariant(E)
 
Endomorphism ring: End(E)\mathrm{End}(E) = Z\Z
Geometric endomorphism ring: End(EQ)\mathrm{End}(E_{\overline{\Q}})  =  Z\Z    (no potential complex multiplication)
sage: E.has_cm()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: ST(E)\mathrm{ST}(E) = SU(2)\mathrm{SU}(2)
Faltings height: hFaltingsh_{\mathrm{Faltings}} ≈ 1.61644622870003090409409639521.6164462287000309040940963952
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 0.811727272482980716793716728590.81172727248298071679371672859
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
abcabc quality: QQ ≈ 1.11076564380244821.1107656438024482
Szpiro ratio: σm\sigma_{m} ≈ 5.6447332469886835.644733246988683

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 1 1
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Mordell-Weil rank: rr = 1 1
comment: Rank
 
sage: E.rank()
 
gp: [lower,upper] = ellrank(E)
 
magma: Rank(E);
 
Regulator: Reg(E/Q)\mathrm{Reg}(E/\Q) ≈ 0.254549582511335356870888856200.25454958251133535687088885620
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: Ω\Omega ≈ 0.666763330507731261833064164910.66676333050773126183306416491
comment: Real Period
 
sage: E.period_lattice().omega()
 
gp: if(E.disc>0,2,1)*E.omega[1]
 
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Tamagawa product: pcp\prod_{p}c_p = 144 144  = 3(223)221 3\cdot( 2^{2} \cdot 3 )\cdot2^{2}\cdot1
comment: Tamagawa numbers
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
oscar: tamagawa_numbers(E)
 
Torsion order: #E(Q)tor\#E(\Q)_{\mathrm{tor}} = 22
comment: Torsion order
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
oscar: prod(torsion_structure(E)[1])
 
Special value: L(E,1) L'(E,1) ≈ 6.11007578692597821677515980276.1100757869259782167751598027
comment: Special L-value
 
r = E.rank();
 
E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Analytic order of Ш: Шan{}_{\mathrm{an}}  ≈  11    (rounded)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 

BSD formula

6.110075787L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor210.6667630.254550144226.110075787\displaystyle 6.110075787 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.666763 \cdot 0.254550 \cdot 144}{2^2} \approx 6.110075787

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
 
E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;
 
Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();
 
omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();
 
assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
 
/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */
 
E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;
 
sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);
 
reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);
 
assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);
 

Modular invariants

Modular form   4650.2.a.bp

q+q2+q3+q4+q6+q8+q94q11+q126q13+q162q17+q184q19+O(q20) q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} - 4 q^{11} + q^{12} - 6 q^{13} + q^{16} - 2 q^{17} + q^{18} - 4 q^{19} + O(q^{20}) Copy content Toggle raw display

comment: q-expansion of modular form
 
sage: E.q_eigenform(20)
 
\\ actual modular form, use for small N
 
[mf,F] = mffromell(E)
 
Ser(mfcoefs(mf,20),q)
 
\\ or just the series
 
Ser(ellan(E,20),q)*q
 
magma: ModularForm(E);
 

For more coefficients, see the Downloads section to the right.

Modular degree: 27648
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
Γ0(N) \Gamma_0(N) -optimal: no
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 4 primes pp of bad reduction:

pp Tamagawa number Kodaira symbol Reduction type Root number ordp(N)\mathrm{ord}_p(N) ordp(Δ)\mathrm{ord}_p(\Delta) ordp(den(j))\mathrm{ord}_p(\mathrm{den}(j))
22 33 I3I_{3} split multiplicative -1 1 3 3
33 1212 I12I_{12} split multiplicative -1 1 12 12
55 44 I1I_{1}^{*} additive 1 2 7 1
3131 11 I1I_{1} nonsplit multiplicative 1 1 1 1

comment: Local data
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]
 

Galois representations

The \ell-adic Galois representation has maximal image for all primes \ell except those listed in the table below.

prime \ell mod-\ell image \ell-adic image
22 2B 4.6.0.1

comment: mod p Galois image
 
sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
 
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

gens = [[1, 0, 8, 1], [1, 8, 0, 1], [1, 4, 4, 17], [3256, 1403, 2329, 2358], [2972, 3719, 2209, 3714], [3484, 1, 503, 6], [7, 6, 3714, 3715], [2328, 473, 2353, 2400], [3713, 8, 3712, 9], [1241, 8, 1244, 33]]
 
GL(2,Integers(3720)).subgroup(gens)
 
Gens := [[1, 0, 8, 1], [1, 8, 0, 1], [1, 4, 4, 17], [3256, 1403, 2329, 2358], [2972, 3719, 2209, 3714], [3484, 1, 503, 6], [7, 6, 3714, 3715], [2328, 473, 2353, 2400], [3713, 8, 3712, 9], [1241, 8, 1244, 33]];
 
sub<GL(2,Integers(3720))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 3720=233531 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 , index 4848, genus 00, and generators

(1081),(1801),(14417),(3256140323292358),(2972371922093714),(348415036),(7637143715),(232847323532400),(3713837129),(12418124433)\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 3256 & 1403 \\ 2329 & 2358 \end{array}\right),\left(\begin{array}{rr} 2972 & 3719 \\ 2209 & 3714 \end{array}\right),\left(\begin{array}{rr} 3484 & 1 \\ 503 & 6 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 3714 & 3715 \end{array}\right),\left(\begin{array}{rr} 2328 & 473 \\ 2353 & 2400 \end{array}\right),\left(\begin{array}{rr} 3713 & 8 \\ 3712 & 9 \end{array}\right),\left(\begin{array}{rr} 1241 & 8 \\ 1244 & 33 \end{array}\right).

Input positive integer mm to see the generators of the reduction of HH to GL2(Z/mZ)\mathrm{GL}_2(\Z/m\Z):

The torsion field K:=Q(E[3720])K:=\Q(E[3720]) is a degree-658243584000658243584000 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/3720Z)\GL_2(\Z/3720\Z).

The table below list all primes \ell for which the Serre invariants associated to the mod-\ell Galois representation are exceptional.

\ell Reduction type Serre weight Serre conductor
22 split multiplicative 44 775=5231 775 = 5^{2} \cdot 31
33 split multiplicative 44 775=5231 775 = 5^{2} \cdot 31
55 additive 1818 186=2331 186 = 2 \cdot 3 \cdot 31
3131 nonsplit multiplicative 3232 150=2352 150 = 2 \cdot 3 \cdot 5^{2}

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree dd for d=d= 2 and 4.
Its isogeny class 4650bi consists of 4 curves linked by isogenies of degrees dividing 4.

Twists

The minimal quadratic twist of this elliptic curve is 930a3, its twist by 55.

Growth of torsion in number fields

The number fields KK of degree less than 24 such that E(K)torsE(K)_{\rm tors} is strictly larger than E(Q)torsE(\Q)_{\rm tors} Z/2Z\cong \Z/{2}\Z are as follows:

[K:Q][K:\Q] KK E(K)torsE(K)_{\rm tors} Base change curve
22 Q(310)\Q(\sqrt{310}) Z/2ZZ/2Z\Z/2\Z \oplus \Z/2\Z not in database
22 Q(31)\Q(\sqrt{31}) Z/4Z\Z/4\Z not in database
22 Q(10)\Q(\sqrt{10}) Z/4Z\Z/4\Z not in database
44 Q(10,31)\Q(\sqrt{10}, \sqrt{31}) Z/2ZZ/4Z\Z/2\Z \oplus \Z/4\Z not in database
88 deg 8 Z/2ZZ/4Z\Z/2\Z \oplus \Z/4\Z not in database
88 deg 8 Z/8Z\Z/8\Z not in database
88 deg 8 Z/8Z\Z/8\Z not in database
88 8.2.31558444171875.2 Z/6Z\Z/6\Z not in database
1616 deg 16 Z/4ZZ/4Z\Z/4\Z \oplus \Z/4\Z not in database
1616 deg 16 Z/2ZZ/8Z\Z/2\Z \oplus \Z/8\Z not in database
1616 deg 16 Z/2ZZ/8Z\Z/2\Z \oplus \Z/8\Z not in database
1616 deg 16 Z/2ZZ/6Z\Z/2\Z \oplus \Z/6\Z not in database
1616 deg 16 Z/12Z\Z/12\Z not in database
1616 deg 16 Z/12Z\Z/12\Z not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.

Iwasawa invariants

pp 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type split split add ss ord ord ord ord ord ord nonsplit ord ord ord ss
λ\lambda-invariant(s) 9 2 - 1,1 1 1 1 1 1 1 1 1 1 1 1,1
μ\mu-invariant(s) 0 0 - 0,0 0 0 0 0 0 0 0 0 0 0 0,0

An entry - indicates that the invariants are not computed because the reduction is additive.

pp-adic regulators

pp-adic regulators are not yet computed for curves that are not Γ0\Gamma_0-optimal.