Properties

 Label 57.a1 Conductor $57$ Discriminant $-171$ j-invariant $$-\frac{1404928}{171}$$ CM no Rank $1$ Torsion structure trivial

Related objects

Show commands: Magma / Pari/GP / SageMath

This is a model for the quotient of the modular curve $X_0(57)$ by its group of Atkin-Lehner involutions $\langle w_3, w_{19} \rangle$.

Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 1, -2, 2])

gp: E = ellinit([0, -1, 1, -2, 2])

magma: E := EllipticCurve([0, -1, 1, -2, 2]);

$$y^2+y=x^3-x^2-2x+2$$

Mordell-Weil group structure

$\Z$

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(2, 1\right)$$ $\hat{h}(P)$ ≈ $0.037574592736823752824805111217$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-1, 1\right)$$, $$\left(-1, -2\right)$$, $$\left(0, 1\right)$$, $$\left(0, -2\right)$$, $$\left(1, 0\right)$$, $$\left(1, -1\right)$$, $$\left(2, 1\right)$$, $$\left(2, -2\right)$$, $$\left(4, 6\right)$$, $$\left(4, -7\right)$$, $$\left(11, 34\right)$$, $$\left(11, -35\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$57$$ = $3 \cdot 19$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-171$ = $-1 \cdot 3^{2} \cdot 19$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{1404928}{171}$$ = $-1 \cdot 2^{12} \cdot 3^{-2} \cdot 7^{3} \cdot 19^{-1}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.83654840927857264432908623559\dots$ Stable Faltings height: $-0.83654840927857264432908623559\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.037574592736823752824805111217\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $5.5555045213725043221174270349\dots$ 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: $2$  = $2\cdot1$ 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: $1$ (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: $L'(E,1)$ ≈ $0.41749163967630964038932049159583966692$

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 4 $\Gamma_0(N)$-optimal: yes Manin constant: 1

Local data

This elliptic curve is semistable. There are 2 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$(j)_{-}$
$3$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$19$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

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 $\ell$.

$p$-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 ss nonsplit ordinary ordinary ordinary ordinary ordinary nonsplit ordinary ordinary ordinary ss ss ordinary ordinary 1,4 1 5 1 1 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 0

Isogenies

This curve has no rational isogenies. Its isogeny class 57.a consists of this curve only.

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 $K$ $3$ 3.1.76.1 $$\Z/2\Z$$ Not in database $6$ 6.0.109744.2 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ 8.2.23085974187.1 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ 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.