Properties

 Label 56b2 Conductor $56$ Discriminant $100352$ j-invariant $$\frac{3543122}{49}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Simplified equation

 $$y^2=x^3-x^2-40x-84$$ y^2=x^3-x^2-40x-84 (homogenize, simplify) $$y^2z=x^3-x^2z-40xz^2-84z^3$$ y^2z=x^3-x^2z-40xz^2-84z^3 (dehomogenize, simplify) $$y^2=x^3-3267x-71010$$ y^2=x^3-3267x-71010 (homogenize, minimize)

sage: E = EllipticCurve([0, -1, 0, -40, -84])

gp: E = ellinit([0, -1, 0, -40, -84])

magma: E := EllipticCurve([0, -1, 0, -40, -84]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

Mordell-Weil group structure

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

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-3, 0\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-3, 0\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$56$$ = $2^{3} \cdot 7$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $100352$ = $2^{11} \cdot 7^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{3543122}{49}$$ = $2 \cdot 7^{-2} \cdot 11^{6}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.23466883556014264871132483940\dots$ Stable Faltings height: $-0.87005375107342584901045428407\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ 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: $1.8960382312277367628312865259\dots$ 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: $2$  = $1\cdot2$ 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: $L(E,1)$ ≈ $0.94801911561386838141564326296$

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);

$$q + 2 q^{3} - 4 q^{5} + q^{7} + q^{9} - 8 q^{15} - 2 q^{17} - 2 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 8 $\Gamma_0(N)$-optimal: no Manin constant: 1

Local data

This elliptic curve is not semistable. There are 2 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 3 11 0
$7$ $2$ $I_{2}$ Split multiplicative -1 1 2 2

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 8.6.0.6
sage: gens = [[1, 2, 2, 5], [17, 4, 34, 9], [2, 1, 27, 0], [3, 4, 8, 11], [9, 50, 48, 7], [53, 4, 52, 5], [1, 4, 0, 1], [1, 0, 4, 1]]

sage: GL(2,Integers(56)).subgroup(gens)

magma: Gens := [[1, 2, 2, 5], [17, 4, 34, 9], [2, 1, 27, 0], [3, 4, 8, 11], [9, 50, 48, 7], [53, 4, 52, 5], [1, 4, 0, 1], [1, 0, 4, 1]];

magma: sub<GL(2,Integers(56))|Gens>;

The image of the adelic Galois representation has level $56$, index $12$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 17 & 4 \\ 34 & 9 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 27 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 9 & 50 \\ 48 & 7 \end{array}\right),\left(\begin{array}{rr} 53 & 4 \\ 52 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$

$p$-adic regulators

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

All $p$-adic regulators are identically $1$ since the rank is $0$.

Iwasawa invariants

$p$ Reduction type 2 7 add split - 3 - 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

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 56b 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 $K$ $2$ $$\Q(\sqrt{2})$$ $$\Z/2\Z \oplus \Z/2\Z$$ 2.2.8.1-392.1-c3 $4$ 4.0.1568.1 $$\Z/4\Z$$ Not in database $8$ 8.4.205520896.2 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.157351936.3 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.2.5377010688.1 $$\Z/6\Z$$ 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.