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Properties

Label 256.a1
Conductor $256$
Discriminant $32768$
j-invariant \( 8000 \)
CM yes (\(D=-8\))
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3+x^2-13x-21\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3+x^2z-13xz^2-21z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-1080x-12096\) Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([0, 1, 0, -13, -21])
 
Copy content gp:E = ellinit([0, 1, 0, -13, -21])
 
Copy content magma:E := EllipticCurve([0, 1, 0, -13, -21]);
 
Copy content oscar:E = elliptic_curve([0, 1, 0, -13, -21])
 
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

\(\Z \oplus \Z/{2}\Z\)

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

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$(5, 8)$$0.96025159501659529387304053183$$\infty$
$(-3, 0)$$0$$2$

Integral points

\( \left(-3, 0\right) \), \((-2,\pm 1)\), \((5,\pm 8)\) Copy content Toggle raw display

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

Invariants

Conductor: $N$  =  \( 256 \) = $2^{8}$
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: $\Delta$  =  $32768$ = $2^{15} $
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: $j$  =  \( 8000 \) = $2^{6} \cdot 5^{3}$
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: $\mathrm{End}(E)$ = $\Z$
Geometric endomorphism ring: $\mathrm{End}(E_{\overline{\Q}})$  =  \(\Z[\sqrt{-2}]\)    (potential complex multiplication)
Copy content comment:Potential complex multiplication
 
Copy content sage:E.has_cm()
 
Copy content magma:HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{ST}(E)$ = $N(\mathrm{U}(1))$
Faltings height: $h_{\mathrm{Faltings}}$ ≈ $-0.40397265035989411975627760561$
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: $h_{\mathrm{stable}}$ ≈ $-1.2704066260598257565278177574$
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
$abc$ quality: $Q$ ≈ $0.9029767420170889$
Szpiro ratio: $\sigma_{m}$ ≈ $3.495723035582761$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$ = $ 1$
Copy content comment:Analytic rank
 
Copy content sage:E.analytic_rank()
 
Copy content gp:ellanalyticrank(E)
 
Copy content magma:AnalyticRank(E);
 
Mordell-Weil rank: $r$ = $ 1$
Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 
Regulator: $\mathrm{Reg}(E/\Q)$ ≈ $0.96025159501659529387304053183$
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: $\Omega$ ≈ $2.5189270468096534385807611190$
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: $\prod_{p}c_p$ = $ 2 $  = $ 2 $
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: $\#E(\Q)_{\mathrm{tor}}$ = $2$
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)$ ≈ $1.2094018572147058551904833617 $
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: Ш${}_{\mathrm{an}}$  ≈  $1$    (rounded)
Copy content comment:Order of Sha
 
Copy content sage:E.sha().an_numerical()
 
Copy content magma:MordellWeilShaInformation(E);
 

BSD formula

$$\begin{aligned} 1.209401857 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.518927 \cdot 0.960252 \cdot 2}{2^2} \\ & \approx 1.209401857\end{aligned}$$

Copy content comment:BSD formula
 
Copy content sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha) E = EllipticCurve([0, 1, 0, -13, -21]); 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([0, 1, 0, -13, -21]); 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   256.2.a.a

\( q - 2 q^{3} + q^{9} - 6 q^{11} - 6 q^{17} - 2 q^{19} + O(q^{20}) \) 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: 16
Copy content comment:Modular degree
 
Copy content sage:E.modular_degree()
 
Copy content gp:ellmoddegree(E)
 
Copy content magma:ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: no
Manin constant: 1
Copy content comment:Manin constant
 
Copy content magma:ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is not semistable. There is only one prime $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $\mathrm{ord}_p(N)$ $\mathrm{ord}_p(\Delta)$ $\mathrm{ord}_p(\mathrm{den}(j))$
$2$ $2$ $III^{*}$ additive 1 8 15 0

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
$2$ 2B 16.192.5.624

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)];
 

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
$2$ additive $4$ \( 1 \)

Isogenies

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

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 256.a consists of 2 curves linked by isogenies of degree 2.

Twists

The minimal quadratic twist of this elliptic curve is 256.a2, its twist by $-8$.

Growth of torsion in number fields

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

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{2}) \) \(\Z/2\Z \oplus \Z/2\Z\) 2.2.8.1-1024.1-d3
$4$ 4.0.2048.2 \(\Z/2\Z \oplus \Z/4\Z\) not in database
$4$ 4.2.18432.2 \(\Z/6\Z\) not in database
$4$ 4.0.6144.1 \(\Z/6\Z\) not in database
$8$ 8.4.67108864.1 \(\Z/2\Z \oplus \Z/4\Z\) not in database
$8$ 8.0.339738624.10 \(\Z/3\Z \oplus \Z/6\Z\) not in database
$8$ 8.4.1358954496.3 \(\Z/2\Z \oplus \Z/6\Z\) not in database
$8$ 8.0.150994944.2 \(\Z/2\Z \oplus \Z/6\Z\) not in database
$12$ 12.0.169075682574336.4 \(\Z/18\Z\) not in database
$16$ 16.0.18014398509481984.1 \(\Z/4\Z \oplus \Z/8\Z\) not in database
$16$ 16.4.4611686018427387904.1 \(\Z/2\Z \oplus \Z/8\Z\) not in database
$16$ 16.0.1846757322198614016.7 \(\Z/6\Z \oplus \Z/6\Z\) not in database
$16$ 16.0.29548117155177824256.5 \(\Z/2\Z \oplus \Z/12\Z\) not in database
$16$ 16.0.364791569817010176.1 \(\Z/2\Z \oplus \Z/12\Z\) not in database
$20$ 20.0.84954018740373771557797888.2 \(\Z/22\Z\) not in database

We only show fields where the torsion growth is primitive.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type add ord ss ss ord ss ord ord ss ss ss ss ord ord ss
$\lambda$-invariant(s) - 3 1,1 3,1 1 1,1 1 1 1,1 1,1 1,1 1,1 1 1 3,1
$\mu$-invariant(s) - 0 0,0 0,0 0 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.

$p$-adic regulators

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.