Properties

Label 15a3
Conductor 1515
Discriminant 225225
j-invariant 13997521225 \frac{13997521}{225}
CM no
Rank 00
Torsion structure Z/2ZZ/4Z\Z/{2}\Z \oplus \Z/{4}\Z

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2+xy+y=x3+x25x+2y^2+xy+y=x^3+x^2-5x+2 Copy content Toggle raw display (homogenize, simplify)
y2z+xyz+yz2=x3+x2z5xz2+2z3y^2z+xyz+yz^2=x^3+x^2z-5xz^2+2z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x36507x+199206y^2=x^3-6507x+199206 Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([1, 1, 1, -5, 2])
 
Copy content gp:E = ellinit([1, 1, 1, -5, 2])
 
Copy content magma:E := EllipticCurve([1, 1, 1, -5, 2]);
 
Copy content oscar:E = elliptic_curve([1, 1, 1, -5, 2])
 
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/2ZZ/4Z\Z/{2}\Z \oplus \Z/{4}\Z

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

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(3,1)(-3, 1)0022
(2,1)(2, 1)0044

Integral points

(3,1) \left(-3, 1\right) , (0,1) \left(0, 1\right) , (0,2) \left(0, -2\right) , (1,1) \left(1, -1\right) , (2,1) \left(2, 1\right) , (2,4) \left(2, -4\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  =  15 15  = 353 \cdot 5
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  =  225225 = 32523^{2} \cdot 5^{2}
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  =  13997521225 \frac{13997521}{225}  = 325224133^{-2} \cdot 5^{-2} \cdot 241^{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}} ≈ 0.74885116236281644215221255789-0.74885116236281644215221255789
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}} ≈ 0.74885116236281644215221255789-0.74885116236281644215221255789
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
abcabc quality: QQ ≈ 0.96229544108906240.9622954410890624
Szpiro ratio: σm\sigma_{m} ≈ 6.0761025751202196.076102575120219

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 ≈ 5.60241216933040809272072334725.6024121693304080927207233472
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  = 22 2\cdot2
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}} = 88
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) ≈ 0.350150760583150505795045209200.35015076058315050579504520920
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}}  =  11    (exact)
Copy content comment:Order of Sha
 
Copy content sage:E.sha().an_numerical()
 
Copy content magma:MordellWeilShaInformation(E);
 

BSD formula

0.350150761L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor215.6024121.0000004820.350150761\begin{aligned} 0.350150761 \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 5.602412 \cdot 1.000000 \cdot 4}{8^2} \\ & \approx 0.350150761\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, 1, 1, -5, 2]); 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, 1, 1, -5, 2]); 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   15.2.a.a

qq2q3q4+q5+q6+3q8+q9q104q11+q122q13q15q16+2q17q18+4q19+O(q20) q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + 3 q^{8} + q^{9} - q^{10} - 4 q^{11} + q^{12} - 2 q^{13} - q^{15} - q^{16} + 2 q^{17} - 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: 2
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: no
Manin constant: 2
Copy content comment:Manin constant
 
Copy content magma:ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is semistable. There are 2 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))
33 22 I2I_{2} nonsplit multiplicative 1 1 2 2
55 22 I2I_{2} split multiplicative -1 1 2 2

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 2Cs 16.96.0.7

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 = [[1, 0, 16, 1], [161, 16, 6, 97], [1, 16, 0, 1], [5, 16, 4, 193], [61, 8, 170, 197], [1, 16, 4, 65], [225, 16, 224, 17], [15, 8, 208, 209]] GL(2,Integers(240)).subgroup(gens)
 
Copy content magma:Gens := [[1, 0, 16, 1], [161, 16, 6, 97], [1, 16, 0, 1], [5, 16, 4, 193], [61, 8, 170, 197], [1, 16, 4, 65], [225, 16, 224, 17], [15, 8, 208, 209]]; sub<GL(2,Integers(240))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 240=2435 240 = 2^{4} \cdot 3 \cdot 5 , index 768768, genus 1313, and generators

(10161),(16116697),(11601),(5164193),(618170197),(116465),(2251622417),(158208209)\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 161 & 16 \\ 6 & 97 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 16 \\ 4 & 193 \end{array}\right),\left(\begin{array}{rr} 61 & 8 \\ 170 & 197 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 4 & 65 \end{array}\right),\left(\begin{array}{rr} 225 & 16 \\ 224 & 17 \end{array}\right),\left(\begin{array}{rr} 15 & 8 \\ 208 & 209 \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[240])K:=\Q(E[240]) is a degree-737280737280 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/240Z)\GL_2(\Z/240\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 good 22 1 1
33 nonsplit multiplicative 44 5 5
55 split multiplicative 66 3 3

Isogenies

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

This curve has non-trivial cyclic isogenies of degree dd for d=d= 2, 4 and 8.
Its isogeny class 15a consists of 8 curves linked by isogenies of degrees dividing 16.

Twists

This elliptic curve is its own minimal quadratic twist.

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/2ZZ/4Z\cong \Z/{2}\Z \oplus \Z/{4}\Z are as follows:

[K:Q][K:\Q] KK E(K)torsE(K)_{\rm tors} Base change curve
22 Q(5)\Q(\sqrt{5}) Z/2ZZ/8Z\Z/2\Z \oplus \Z/8\Z 2.2.5.1-45.1-a6
44 Q(i,15)\Q(i, \sqrt{15}) Z/4ZZ/4Z\Z/4\Z \oplus \Z/4\Z not in database
44 Q(ζ12)\Q(\zeta_{12}) Z/2ZZ/8Z\Z/2\Z \oplus \Z/8\Z not in database
44 Q(ζ15)+\Q(\zeta_{15})^+ Z/2ZZ/16Z\Z/2\Z \oplus \Z/16\Z not in database
88 8.0.12960000.1 Z/4ZZ/8Z\Z/4\Z \oplus \Z/8\Z not in database
88 Q(ζ20)\Q(\zeta_{20}) Z/2ZZ/16Z\Z/2\Z \oplus \Z/16\Z not in database
88 8.2.110716875.2 Z/2ZZ/12Z\Z/2\Z \oplus \Z/12\Z not in database
1616 deg 16 Z/4ZZ/8Z\Z/4\Z \oplus \Z/8\Z not in database
1616 deg 16 Z/4ZZ/8Z\Z/4\Z \oplus \Z/8\Z not in database
1616 16.0.1426576071720960000.2 Z/2ZZ/16Z\Z/2\Z \oplus \Z/16\Z not in database
1616 Q(ζ60)\Q(\zeta_{60}) Z/4ZZ/16Z\Z/4\Z \oplus \Z/16\Z not in database
1616 deg 16 Z/2ZZ/32Z\Z/2\Z \oplus \Z/32\Z not in database
1616 deg 16 Z/2ZZ/24Z\Z/2\Z \oplus \Z/24\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
Reduction type ord nonsplit split
λ\lambda-invariant(s) 0 0 1
μ\mu-invariant(s) 0 0 0

All Iwasawa λ\lambda and μ\mu-invariants for primes p3p\ge 3 of good reduction are zero.

pp-adic regulators

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