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This is a model for the quotient of the modular curve $X_0(61)$ by its Atkin-Lehner involution $w_{61}$.

## Minimal Weierstrass equation

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

sage: E = EllipticCurve("61a1")

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

gp: E = ellinit("61a1")

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

magma: E := EllipticCurve("61a1");

$$y^2 + x y = x^{3} - 2 x + 1$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(1, -1\right)$$ $$\hat{h}(P)$$ ≈ 0.07918773136204195

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-1, 2\right)$$, $$\left(-1, -1\right)$$, $$\left(0, 1\right)$$, $$\left(0, -1\right)$$, $$\left(1, 0\right)$$, $$\left(1, -1\right)$$, $$\left(7, 15\right)$$, $$\left(7, -22\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$61$$ = $$61$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-61$$ = $$-1 \cdot 61$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{912673}{61}$$ = $$-1 \cdot 61^{-1} \cdot 97^{3}$$ 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: $$0.079187731362$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$6.13319314839$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$1$$  = $$1$$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $$1$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form61.2.a.a

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2 $$\Gamma_0(N)$$-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/factorial(ar)

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

$$L'(E,1)$$ ≈ $$0.485673651427$$

## Local data

This elliptic curve is semistable.

sage: E.local_data()

gp: ellglobalred(E)

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

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$61$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1

## Galois representations

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

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 $$p$$ .

## $p$-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 61 ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary nonsplit 1 1 1 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

## Isogenies

This curve has no rational isogenies. Its isogeny class 61a 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 3.1.244.1 $$\Z/2\Z$$ Not in database
6 6.0.14526784.1 $$\Z/2\Z \times \Z/2\Z$$ 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.