# Properties

 Label 182070cj6 Conductor 182070 Discriminant -4736305493311157226562500 j-invariant $$\frac{226523624554079}{269165039062500}$$ CM no Rank 2 Torsion Structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, 3303216, 104681219188]) # or

sage: E = EllipticCurve("182070cj6")

gp: E = ellinit([1, -1, 0, 3303216, 104681219188]) \\ or

gp: E = ellinit("182070cj6")

magma: E := EllipticCurve([1, -1, 0, 3303216, 104681219188]); // or

magma: E := EllipticCurve("182070cj6");

$$y^2 + x y = x^{3} - x^{2} + 3303216 x + 104681219188$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(40677, -8238776\right)$$ $$\left(302, -325276\right)$$ $$\hat{h}(P)$$ ≈ 2.051719284736134 0.7210090561550054

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-\frac{17917}{4}, \frac{17917}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-3073, 257474\right)$$, $$\left(-3073, -254401\right)$$, $$\left(-2758, 274484\right)$$, $$\left(-2758, -271726\right)$$, $$\left(-573, 320599\right)$$, $$\left(-573, -320026\right)$$, $$\left(302, 324974\right)$$, $$\left(302, -325276\right)$$, $$\left(3192, 382774\right)$$, $$\left(3192, -385966\right)$$, $$\left(8802, 898724\right)$$, $$\left(8802, -907526\right)$$, $$\left(11777, 1327124\right)$$, $$\left(11777, -1338901\right)$$, $$\left(40677, 8198099\right)$$, $$\left(40677, -8238776\right)$$, $$\left(56777, 13511099\right)$$, $$\left(56777, -13567876\right)$$, $$\left(2826552, 4750685974\right)$$, $$\left(2826552, -4753512526\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$182070$$ = $$2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-4736305493311157226562500$$ = $$-1 \cdot 2^{2} \cdot 3^{8} \cdot 5^{16} \cdot 7^{2} \cdot 17^{6}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{226523624554079}{269165039062500}$$ = $$2^{-2} \cdot 3^{-2} \cdot 5^{-16} \cdot 7^{-2} \cdot 47^{3} \cdot 1297^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1.39300891207$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.0603384713115$$ 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: $$512$$  = $$2\cdot2\cdot2^{4}\cdot2\cdot2^{2}$$ 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: $$1$$ (rounded)

## Modular invariants

#### Modular form 182070.2.a.bi

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);

$$q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} - q^{10} - 4q^{11} - 2q^{13} - q^{14} + q^{16} + 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 41943040 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

#### Special L-value

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);

$$L^{(2)}(E,1)/2!$$ ≈ $$10.7586596195$$

## Local data

This elliptic curve is not semistable.

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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$3$$ $$2$$ $$I_2^{*}$$ Additive -1 2 8 2
$$5$$ $$16$$ $$I_{16}$$ Split multiplicative -1 1 16 16
$$7$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$17$$ $$4$$ $$I_0^{*}$$ 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 X207.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^4\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 8 & 3 \end{array}\right),\left(\begin{array}{rr} 7 & 14 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 3 \end{array}\right)$ and has index 48.

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

prime Image of Galois representation
$$2$$ B

## $p$-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.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4 and 8.
Its isogeny class 182070cj consists of 8 curves linked by isogenies of degrees dividing 16.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{-1})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
$$\Q(\sqrt{-51})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{51})$$ $$\Z/4\Z$$ Not in database
4 $$\Q(i, \sqrt{51})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{-6}, \sqrt{34})$$ $$\Z/8\Z$$ Not in database
$$\Q(\sqrt{6}, \sqrt{-34})$$ $$\Z/8\Z$$ Not in database
$$\Q(\sqrt{14}, \sqrt{51})$$ $$\Z/8\Z$$ Not in database
$$\Q(\sqrt{-14}, \sqrt{51})$$ $$\Z/8\Z$$ 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.