# Properties

 Label 55470.y2 Conductor $55470$ Discriminant $-269584200$ j-invariant $$\frac{222641831}{145800}$$ CM no Rank $2$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, 155, -205]) # or

sage: E = EllipticCurve("55470.y2")

gp: E = ellinit([1, 1, 1, 155, -205]) \\ or

gp: E = ellinit("55470.y2")

magma: E := EllipticCurve([1, 1, 1, 155, -205]); // or

magma: E := EllipticCurve("55470.y2");

$$y^2+xy+y=x^3+x^2+155x-205$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(5, 24\right)$$ $$\left(13, 58\right)$$ $$\hat{h}(P)$$ ≈ $0.61008656517874116175958946283$ $1.8046838556792165698301309098$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(5, 24\right)$$, $$\left(5, -30\right)$$, $$\left(9, 40\right)$$, $$\left(9, -50\right)$$, $$\left(13, 58\right)$$, $$\left(13, -72\right)$$, $$\left(113, 1158\right)$$, $$\left(113, -1272\right)$$, $$\left(733, 19498\right)$$, $$\left(733, -20232\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$55470$$ = $$2 \cdot 3 \cdot 5 \cdot 43^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-269584200$$ = $$-1 \cdot 2^{3} \cdot 3^{6} \cdot 5^{2} \cdot 43^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{222641831}{145800}$$ = $$2^{-3} \cdot 3^{-6} \cdot 5^{-2} \cdot 43 \cdot 173^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1.0262642412107110450847271377$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.99385138914273995823172137245$$ 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: $$12$$  = $$3\cdot2\cdot2\cdot1$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$1$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

Modular form 55470.2.a.y

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);

$$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - 2q^{7} + q^{8} + q^{9} + q^{10} - 3q^{11} - q^{12} - 4q^{13} - 2q^{14} - q^{15} + q^{16} - 3q^{17} + q^{18} + q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 24192 $$\Gamma_0(N)$$-optimal: yes 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);

$$L^{(2)}(E,1)/2!$$ ≈ $$12.239449701057421545122685728423252977$$

## Local data

This elliptic curve is semistable. There are 4 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$$(j)_{-}$$
$$2$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$3$$ $$2$$ $$I_{6}$$ Non-split multiplicative 1 1 6 6
$$5$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$43$$ $$1$$ $$II$$ Additive -1 2 2 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 X4.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 7 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 1 & 1 \end{array}\right)$ and has index 2.

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
$$3$$ B

## $p$-adic data

### $p$-adic regulators

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

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 split nonsplit split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary add ordinary 6 2 3 2 2 2 2 2 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 $$d=$$ 3.
Its isogeny class 55470.y consists of 2 curves linked by isogenies of degree 3.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-43})$$ $$\Z/3\Z$$ Not in database $3$ 3.1.14792.1 $$\Z/2\Z$$ Not in database $6$ 6.0.1750426112.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.2.2480767475625.1 $$\Z/3\Z$$ Not in database $6$ 6.0.9408540352.1 $$\Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ Deg 12 $$\Z/3\Z \times \Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.0.255519771233789729519436195574238766891796875.1 $$\Z/9\Z$$ Not in database $18$ 18.2.4002193664640074322781395499584000000000000.2 $$\Z/6\Z$$ 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.