Properties

 Label 1520.b1 Conductor $1520$ Discriminant $6516050000$ j-invariant $$\frac{5405726654464}{407253125}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Simplified equation

 $$y^2=x^3+x^2-921x-10346$$ y^2=x^3+x^2-921x-10346 (homogenize, simplify) $$y^2z=x^3+x^2z-921xz^2-10346z^3$$ y^2z=x^3+x^2z-921xz^2-10346z^3 (dehomogenize, simplify) $$y^2=x^3-74628x-7318377$$ y^2=x^3-74628x-7318377 (homogenize, minimize)

sage: E = EllipticCurve([0, 1, 0, -921, -10346])

gp: E = ellinit([0, 1, 0, -921, -10346])

magma: E := EllipticCurve([0, 1, 0, -921, -10346]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

Mordell-Weil group structure

$$\Z \oplus \Z/{2}\Z$$

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(62, 418\right)$$ (62, 418) $\hat{h}(P)$ ≈ $2.1688769544000068094200299101$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-14, 0\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-14, 0\right)$$, $$(62,\pm 418)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$1520$$ = $2^{4} \cdot 5 \cdot 19$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $6516050000$ = $2^{4} \cdot 5^{5} \cdot 19^{4}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{5405726654464}{407253125}$$ = $2^{14} \cdot 5^{-5} \cdot 19^{-4} \cdot 691^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.62837344719942831334375604364\dots$ Stable Faltings height: $0.39732438701277987687134533649\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $2.1688769544000068094200299101\dots$ 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); Real period: $0.87068173419410979797942301164\dots$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $4$  = $1\cdot1\cdot2^{2}$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $2$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (exact) sage: r = E.rank(); sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: ar = ellanalyticrank(E); gp: ar[2]/factorial(ar[1])  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); Special value: $L'(E,1)$ ≈ $1.8884015479106371258984507536$

Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

magma: ModularForm(E);

$$q - 2 q^{3} - q^{5} - 2 q^{7} + q^{9} + 6 q^{13} + 2 q^{15} + 2 q^{17} + q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 960 $\Gamma_0(N)$-optimal: yes Manin constant: 1

Local data

This elliptic curve is not semistable. There are 3 primes of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $1$ $II$ Additive -1 4 4 0
$5$ $1$ $I_{5}$ Non-split multiplicative 1 1 5 5
$19$ $4$ $I_{4}$ Split multiplicative -1 1 4 4

Galois representations

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

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 4.6.0.3

The image of the adelic Galois representation has level $760$, index $48$, and genus $0$.

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

Iwasawa invariants

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

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 1520.b consists of 2 curves linked by isogenies of degree 2.

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:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{5})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $4$ 4.0.320.1 $$\Z/4\Z$$ Not in database $8$ 8.4.64000000.1 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.2560000.1 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/6\Z$$ Not in database $16$ 16.0.4096000000000000.1 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ 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.