Properties

 Label 117117.l3 Conductor 117117 Discriminant 9200420605705329 j-invariant $$\frac{6570725617}{2614689}$$ CM no Rank 2 Torsion Structure $$\Z/{2}\Z \times \Z/{2}\Z$$

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, -59351, -3095634]); // or

magma: E := EllipticCurve("117117r2");

sage: E = EllipticCurve([1, -1, 1, -59351, -3095634]) # or

sage: E = EllipticCurve("117117r2")

gp: E = ellinit([1, -1, 1, -59351, -3095634]) \\ or

gp: E = ellinit("117117r2")

$$y^2 + x y + y = x^{3} - x^{2} - 59351 x - 3095634$$

Mordell-Weil group structure

$$\Z^2 \times \Z/{2}\Z \times \Z/{2}\Z$$

Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-202, 888\right)$$ $$\left(-198, 1028\right)$$ $$\hat{h}(P)$$ ≈ 4.50090410412 1.83114799394

Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(-55, 27\right)$$, $$\left(\frac{1067}{4}, -\frac{1071}{8}\right)$$

Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-211, 105\right)$$, $$\left(-202, 888\right)$$, $$\left(-202, -687\right)$$, $$\left(-198, 1028\right)$$, $$\left(-198, -831\right)$$, $$\left(-130, 1617\right)$$, $$\left(-130, -1488\right)$$, $$\left(-64, 693\right)$$, $$\left(-64, -630\right)$$, $$\left(-55, 27\right)$$, $$\left(296, 2133\right)$$, $$\left(296, -2430\right)$$, $$\left(377, 5103\right)$$, $$\left(377, -5481\right)$$, $$\left(1389, 50225\right)$$, $$\left(1389, -51615\right)$$, $$\left(1661, 66093\right)$$, $$\left(1661, -67755\right)$$, $$\left(5522, 407148\right)$$, $$\left(5522, -412671\right)$$, $$\left(8070, 720552\right)$$, $$\left(8070, -728623\right)$$

Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$117117$$ = $$3^{2} \cdot 7 \cdot 11 \cdot 13^{2}$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$9200420605705329$$ = $$3^{8} \cdot 7^{4} \cdot 11^{2} \cdot 13^{6}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{6570725617}{2614689}$$ = $$3^{-2} \cdot 7^{-4} \cdot 11^{-2} \cdot 1873^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$2$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$4.81010586828$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.316548815236$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$64$$  = $$2^{2}\cdot2\cdot2\cdot2^{2}$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$4$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (rounded)

Modular invariants

Modular form 117117.2.a.l

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

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

$$q - q^{2} - q^{4} - 2q^{5} - q^{7} + 3q^{8} + 2q^{10} - q^{11} + q^{14} - q^{16} - 2q^{17} - 4q^{19} + O(q^{20})$$

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 614400 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar[2]/factorial(ar[1])

$$L^{(2)}(E,1)/2!$$ ≈ $$6.09053325505$$

Local data

This elliptic curve is not semistable.

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

sage: E.local_data()

gp: ellglobalred(E)[5]

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$3$$ $$4$$ $$I_2^{*}$$ Additive -1 2 8 2
$$7$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$11$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$13$$ $$4$$ $$I_0^{*}$$ Additive 1 2 6 0

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X25.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right)$ and has index 12.

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

sage: rho = E.galois_representation();

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

The mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$2$$ Cs

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

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

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 ordinary add ordinary nonsplit nonsplit add ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary 5 - 2 2 2 - 2 2 2,4 2 2 2 2 2 2 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 and 4.
Its isogeny class 117117.l consists of 6 curves linked by isogenies of degrees dividing 8.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{-39})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
4 $$\Q(\sqrt{33}, \sqrt{39})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{-33}, \sqrt{39})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.