# Properties

 Label 333270eg5 Conductor 333270 Discriminant 19918489084286132812500 j-invariant $$\frac{9150443179640281}{184570312500}$$ CM no Rank 1 Torsion Structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -20746157, -35726285311]) # or

sage: E = EllipticCurve("333270eg5")

gp: E = ellinit([1, -1, 1, -20746157, -35726285311]) \\ or

gp: E = ellinit("333270eg5")

magma: E := EllipticCurve([1, -1, 1, -20746157, -35726285311]); // or

magma: E := EllipticCurve("333270eg5");

$$y^2 + x y + y = x^{3} - x^{2} - 20746157 x - 35726285311$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(79747, 22442626\right)$$ $$\hat{h}(P)$$ ≈ 3.545055953009774

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-\frac{11637}{4}, \frac{11633}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-2853, 16726\right)$$, $$\left(-2853, -13874\right)$$, $$\left(79747, 22442626\right)$$, $$\left(79747, -22522374\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$333270$$ = $$2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$19918489084286132812500$$ = $$2^{2} \cdot 3^{9} \cdot 5^{12} \cdot 7 \cdot 23^{6}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{9150443179640281}{184570312500}$$ = $$2^{-2} \cdot 3^{-3} \cdot 5^{-12} \cdot 7^{-1} \cdot 37^{3} \cdot 5653^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$3.54505595301$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.0708267940177$$ 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: $$192$$  = $$2\cdot2^{2}\cdot( 2^{2} \cdot 3 )\cdot1\cdot2$$ 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)

## Modular invariants

#### Modular form 333270.2.a.eg

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);

$$q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} + q^{10} + 2q^{13} - q^{14} + q^{16} - 6q^{17} - 8q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 38928384 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$12.0520774927$$

## 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$3$$ $$4$$ $$I_3^{*}$$ Additive -1 2 9 3
$$5$$ $$12$$ $$I_{12}$$ Split multiplicative -1 1 12 12
$$7$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$23$$ $$2$$ $$I_0^{*}$$ 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 X13.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$ and has index 6.

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
$$3$$ 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, 3, 4, 6 and 12.
Its isogeny class 333270eg consists of 8 curves linked by isogenies of degrees dividing 12.

## 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 $$\Q(\sqrt{21})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
$$\Q(\sqrt{69})$$ $$\Z/6\Z$$ Not in database
$$\Q(\sqrt{-69})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{-161})$$ $$\Z/4\Z$$ Not in database
4 $$\Q(i, \sqrt{69})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{21}, \sqrt{69})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
$$\Q(\sqrt{21}, \sqrt{-69})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{-21}, \sqrt{69})$$ $$\Z/12\Z$$ Not in database
6 6.0.4206667248.2 $$\Z/6\Z$$ 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.