Properties

Label 438702x1
Conductor 438702438702
Discriminant 1.201×1015-1.201\times 10^{15}
j-invariant 3715939375349741824 -\frac{37159393753}{49741824}
CM no
Rank 00
Torsion structure Z/2Z\Z/{2}\Z

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2+xy+y=x320092x1996870y^2+xy+y=x^3-20092x-1996870 Copy content Toggle raw display (homogenize, simplify)
y2z+xyz+yz2=x320092xz21996870z3y^2z+xyz+yz^2=x^3-20092xz^2-1996870z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x326038611x93087839250y^2=x^3-26038611x-93087839250 Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([1, 0, 1, -20092, -1996870])
 
Copy content gp:E = ellinit([1, 0, 1, -20092, -1996870])
 
Copy content magma:E := EllipticCurve([1, 0, 1, -20092, -1996870]);
 
Copy content oscar:E = elliptic_curve([1, 0, 1, -20092, -1996870])
 
Copy content comment:Simplified equation
 
Copy content sage:E.short_weierstrass_model()
 
Copy content magma:WeierstrassModel(E);
 
Copy content oscar:short_weierstrass_model(E)
 

Mordell-Weil group structure

Z/2Z\Z/{2}\Z

Copy content comment:Mordell-Weil group
 
Copy content magma:MordellWeilGroup(E);
 

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(177,89)(177, -89)0022

Integral points

(177,89) \left(177, -89\right) Copy content Toggle raw display

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

Invariants

Conductor: NN  =  438702 438702  = 2311172232 \cdot 3 \cdot 11 \cdot 17^{2} \cdot 23
Copy content comment:Conductor
 
Copy content sage:E.conductor().factor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Copy content oscar:conductor(E)
 
Discriminant: Δ\Delta  =  1200646708985856-1200646708985856 = 121631117623-1 \cdot 2^{16} \cdot 3 \cdot 11 \cdot 17^{6} \cdot 23
Copy content comment:Discriminant
 
Copy content sage:E.discriminant().factor()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Copy content oscar:discriminant(E)
 
j-invariant: jj  =  3715939375349741824 -\frac{37159393753}{49741824}  = 121631111231473713-1 \cdot 2^{-16} \cdot 3^{-1} \cdot 11^{-1} \cdot 23^{-1} \cdot 47^{3} \cdot 71^{3}
Copy content comment:j-invariant
 
Copy content sage:E.j_invariant().factor()
 
Copy content gp:E.j
 
Copy content magma:jInvariant(E);
 
Copy content 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)
Copy content comment:Potential complex multiplication
 
Copy content sage:E.has_cm()
 
Copy content magma:HasComplexMultiplication(E);
 
Sato-Tate group: ST(E)\mathrm{ST}(E) = SU(2)\mathrm{SU}(2)
Faltings height: hFaltingsh_{\mathrm{Faltings}} ≈ 1.58620391872066383942079880311.5862039187206638394207988031
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 0.169597246692555799296031494160.16959724669255579929603149416
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
abcabc quality: QQ ≈ 1.02197456352252411.0219745635225241
Szpiro ratio: σm\sigma_{m} ≈ 3.27409483271872273.2740948327187227

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 0 0
Copy content comment:Analytic rank
 
Copy content sage:E.analytic_rank()
 
Copy content gp:ellanalyticrank(E)
 
Copy content magma:AnalyticRank(E);
 
Mordell-Weil rank: rr = 0 0
Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 
Regulator: Reg(E/Q)\mathrm{Reg}(E/\Q) = 11
Copy content comment:Regulator
 
Copy content sage:E.regulator()
 
Copy content gp:G = E.gen \\ if available matdet(ellheightmatrix(E,G))
 
Copy content magma:Regulator(E);
 
Real period: Ω\Omega ≈ 0.191048810821507176136511262700.19104881082150717613651126270
Copy content comment:Real Period
 
Copy content sage:E.period_lattice().omega()
 
Copy content gp:if(E.disc>0,2,1)*E.omega[1]
 
Copy content magma:(Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Tamagawa product: pcp\prod_{p}c_p = 4 4  = 21121 2\cdot1\cdot1\cdot2\cdot1
Copy content comment:Tamagawa numbers
 
Copy content sage:E.tamagawa_numbers()
 
Copy content gp:gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Copy content magma:TamagawaNumbers(E);
 
Copy content oscar:tamagawa_numbers(E)
 
Torsion order: #E(Q)tor\#E(\Q)_{\mathrm{tor}} = 22
Copy content comment:Torsion order
 
Copy content sage:E.torsion_order()
 
Copy content gp:elltors(E)[1]
 
Copy content magma:Order(TorsionSubgroup(E));
 
Copy content oscar:prod(torsion_structure(E)[1])
 
Special value: L(E,1) L(E,1) ≈ 0.764195243286028704546045050810.76419524328602870454604505081
Copy content comment:Special L-value
 
Copy content sage:r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
Copy content gp:[r,L1r] = ellanalyticrank(E); L1r/r!
 
Copy content magma:Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Analytic order of Ш: Шan{}_{\mathrm{an}}  =  44 = 222^2    (exact)
Copy content comment:Order of Sha
 
Copy content sage:E.sha().an_numerical()
 
Copy content magma:MordellWeilShaInformation(E);
 

BSD formula

0.764195243L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor240.1910491.0000004220.764195243\begin{aligned} 0.764195243 \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{4 \cdot 0.191049 \cdot 1.000000 \cdot 4}{2^2} \\ & \approx 0.764195243\end{aligned}

Copy content comment:BSD formula
 
Copy content sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha) E = EllipticCurve([1, 0, 1, -20092, -1996870]); 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)))
 
Copy content magma:/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */ E := EllipticCurve([1, 0, 1, -20092, -1996870]); 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 438702.2.a.x

qq2+q3+q42q5q6q8+q9+2q10q11+q122q132q15+q16q184q19+O(q20) q - q^{2} + q^{3} + q^{4} - 2 q^{5} - q^{6} - q^{8} + q^{9} + 2 q^{10} - q^{11} + q^{12} - 2 q^{13} - 2 q^{15} + q^{16} - q^{18} - 4 q^{19} + O(q^{20}) Copy content Toggle raw display

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:\\ 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
 
Copy content magma:ModularForm(E);
 

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

Modular degree: 1769472
Copy content comment:Modular degree
 
Copy content sage:E.modular_degree()
 
Copy content gp:ellmoddegree(E)
 
Copy content magma:ModularDegree(E);
 
Γ0(N) \Gamma_0(N) -optimal: not computed* (one of 3 curves in this isogeny class which might be optimal)
Manin constant: 1 (conditional*)
Copy content comment:Manin constant
 
Copy content magma:ManinConstant(E);
 
* The optimal curve in each isogeny class has not been determined in all cases for conductors over 400000. The Manin constant is correct provided that this curve is optimal.

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 5 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 22 I16I_{16} nonsplit multiplicative 1 1 16 16
33 11 I1I_{1} split multiplicative -1 1 1 1
1111 11 I1I_{1} nonsplit multiplicative 1 1 1 1
1717 22 I0I_0^{*} additive 1 2 6 0
2323 11 I1I_{1} nonsplit multiplicative 1 1 1 1

Copy content comment:Local data
 
Copy content sage:E.local_data()
 
Copy content gp:ellglobalred(E)[5]
 
Copy content magma:[LocalInformation(E,p) : p in BadPrimes(E)];
 
Copy content 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

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

Copy content comment:Adelic image of Galois representation
 
Copy content sage:gens = [[52276, 66793, 91103, 24294], [103217, 8, 103216, 9], [32641, 32640, 29614, 20503], [1, 0, 8, 1], [1, 8, 0, 1], [1, 4, 4, 17], [31468, 66793, 100487, 24294], [30359, 0, 0, 103223], [6835, 6834, 81226, 72115], [20248, 97155, 93109, 66794], [7, 6, 103218, 103219]] GL(2,Integers(103224)).subgroup(gens)
 
Copy content magma:Gens := [[52276, 66793, 91103, 24294], [103217, 8, 103216, 9], [32641, 32640, 29614, 20503], [1, 0, 8, 1], [1, 8, 0, 1], [1, 4, 4, 17], [31468, 66793, 100487, 24294], [30359, 0, 0, 103223], [6835, 6834, 81226, 72115], [20248, 97155, 93109, 66794], [7, 6, 103218, 103219]]; sub<GL(2,Integers(103224))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 103224=233111723 103224 = 2^{3} \cdot 3 \cdot 11 \cdot 17 \cdot 23 , index 4848, genus 00, and generators

(52276667939110324294),(10321781032169),(32641326402961420503),(1081),(1801),(14417),(314686679310048724294),(3035900103223),(683568348122672115),(20248971559310966794),(76103218103219)\left(\begin{array}{rr} 52276 & 66793 \\ 91103 & 24294 \end{array}\right),\left(\begin{array}{rr} 103217 & 8 \\ 103216 & 9 \end{array}\right),\left(\begin{array}{rr} 32641 & 32640 \\ 29614 & 20503 \end{array}\right),\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} 31468 & 66793 \\ 100487 & 24294 \end{array}\right),\left(\begin{array}{rr} 30359 & 0 \\ 0 & 103223 \end{array}\right),\left(\begin{array}{rr} 6835 & 6834 \\ 81226 & 72115 \end{array}\right),\left(\begin{array}{rr} 20248 & 97155 \\ 93109 & 66794 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 103218 & 103219 \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[103224])K:=\Q(E[103224]) is a degree-424337074657689600424337074657689600 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/103224Z)\GL_2(\Z/103224\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 nonsplit multiplicative 44 219351=31117223 219351 = 3 \cdot 11 \cdot 17^{2} \cdot 23
33 split multiplicative 44 146234=21117223 146234 = 2 \cdot 11 \cdot 17^{2} \cdot 23
1111 nonsplit multiplicative 1212 39882=2317223 39882 = 2 \cdot 3 \cdot 17^{2} \cdot 23
1717 additive 146146 1518=231123 1518 = 2 \cdot 3 \cdot 11 \cdot 23
2323 nonsplit multiplicative 2424 19074=2311172 19074 = 2 \cdot 3 \cdot 11 \cdot 17^{2}

Isogenies

Copy content comment:Isogenies
 
Copy content gp:ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree dd for d=d= 2 and 4.
Its isogeny class 438702x consists of 4 curves linked by isogenies of degrees dividing 4.

Twists

The minimal quadratic twist of this elliptic curve is 1518f1, its twist by 1717.

Iwasawa invariants

No Iwasawa invariant data is available for this curve.

pp-adic regulators

All pp-adic regulators are identically 11 since the rank is 00.