# Properties

 Label 5054b1 Conductor $5054$ Discriminant $-4.476\times 10^{17}$ j-invariant $$\frac{53261199}{26353376}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, 20148, 32163887])

gp: E = ellinit([1, -1, 1, 20148, 32163887])

magma: E := EllipticCurve([1, -1, 1, 20148, 32163887]);

$$y^2+xy+y=x^3-x^2+20148x+32163887$$

## Mordell-Weil group structure

$\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(271, 7445\right)$$ (271, 7445) $\hat{h}(P)$ ≈ $0.23428542293602147057813604464$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(145, 6101\right)$$, $$\left(145, -6247\right)$$, $$\left(271, 7445\right)$$, $$\left(271, -7717\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$5054$$ = $2 \cdot 7 \cdot 19^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-447574222639176416$ = $-1 \cdot 2^{5} \cdot 7^{7} \cdot 19^{8}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{53261199}{26353376}$$ = $2^{-5} \cdot 3^{3} \cdot 7^{-7} \cdot 19 \cdot 47^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.0659636738345777293519765893\dots$ Stable Faltings height: $0.10300435439028408934595830138\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.23428542293602147057813604464\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.23104263642920630508424412789\dots$ 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: $105$  = $5\cdot7\cdot3$ 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: $1$ (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: $L'(E,1)$ ≈ $5.6836417881673542659860103321$

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 47880 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 3 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$(j)_{-}$
$2$ $5$ $I_{5}$ Split multiplicative -1 1 5 5
$7$ $7$ $I_{7}$ Split multiplicative -1 1 7 7
$19$ $3$ $IV^{*}$ Additive 1 2 8 0

## 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
$7$ 7B.2.1 7.16.0.1

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 ss ord split ord ord ss add ord ord ord ord ord ord ss 2 3,3 3 2 1 1 1,1 - 1 1 1 1 1 1 1,1 0 0,0 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 $d=$ 7.
Its isogeny class 5054b consists of 2 curves linked by isogenies of degree 7.

## 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 $K$ $3$ 3.1.20216.1 $$\Z/2\Z$$ Not in database $3$ 3.3.361.1 $$\Z/7\Z$$ Not in database $6$ 6.0.22886452736.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ 8.2.10949022029232.4 $$\Z/3\Z$$ Not in database $9$ 9.3.8262009437696.1 $$\Z/14\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $18$ 18.0.11987688643769434377362138464256.1 $$\Z/2\Z \times \Z/14\Z$$ 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.