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

## Minimal Weierstrass equation

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

magma: E := EllipticCurve("57a1");

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

sage: E = EllipticCurve("57a1")

gp: E = ellinit([0, -1, 1, -2, 2]) \\ or

gp: E = ellinit("57a1")

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

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(2, -2\right)$$ $$\hat{h}(P)$$ ≈ 0.0375745927368

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

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

Note: only one of each pair $\pm P$ is listed.

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E) Conductor: $$57$$ = $$3 \cdot 19$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-171$$ = $$-1 \cdot 3^{2} \cdot 19$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{1404928}{171}$$ = $$-1 \cdot 2^{12} \cdot 3^{-2} \cdot 7^{3} \cdot 19^{-1}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$1$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$0.0375745927368$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega Real period: $$5.55550452137$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$2$$  = $$2\cdot1$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E) Torsion order: $$1$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form57.2.a.a

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

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

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

#### Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar/factorial(ar)

$$L'(E,1)$$ ≈ $$0.417491639676$$

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

sage: E.local_data()

gp: ellglobalred(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

The 2-adic representation attached to this elliptic curve is surjective.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

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

## $p$-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 $$p\ge5$$ 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 57a consists of this curve only.

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

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
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

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.