# Properties

 Label 58a1 Conductor $58$ Discriminant $-116$ j-invariant $$-\frac{185193}{116}$$ 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 $X_0(58)$ by its group of Atkin-Lehner involutions $\langle w_2, w_{29} \rangle$.

## Simplified equation

 $$y^2+xy=x^3-x^2-x+1$$ y^2+xy=x^3-x^2-x+1 (homogenize, simplify) $$y^2z+xyz=x^3-x^2z-xz^2+z^3$$ y^2z+xyz=x^3-x^2z-xz^2+z^3 (dehomogenize, simplify) $$y^2=x^3-19x+46$$ y^2=x^3-19x+46 (homogenize, minimize)

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

gp: E = ellinit([1, -1, 0, -1, 1])

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

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(0, 1\right)$$ (0, 1) $\hat{h}(P)$ ≈ $0.042420307839929849766777051945$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-1, 1\right)$$, $$\left(-1, 0\right)$$, $$\left(0, 1\right)$$, $$\left(0, -1\right)$$, $$\left(1, 0\right)$$, $$\left(1, -1\right)$$, $$\left(2, 1\right)$$, $$\left(2, -3\right)$$, $$\left(4, 5\right)$$, $$\left(4, -9\right)$$, $$\left(15, 49\right)$$, $$\left(15, -64\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$58$$ = $2 \cdot 29$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-116$ = $-1 \cdot 2^{2} \cdot 29$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{185193}{116}$$ = $-1 \cdot 2^{-2} \cdot 3^{3} \cdot 19^{3} \cdot 29^{-1}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.90222852034927295517998938458\dots$ Stable Faltings height: $-0.90222852034927295517998938458\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.042420307839929849766777051945\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $5.4655916988863770997253673433\dots$ 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: $L'(E,1)$ ≈ $0.46370416478825057844059634784$

## 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);

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

 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$(j)_{-}$
$2$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$29$ $1$ $I_{1}$ 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 nonsplit ss ord ord ord ord ord ord ss nonsplit ord ord ord ord ord 1 5,1 1 1 1 1 1 1 1,1 1 3 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 58a 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 $K$ $3$ 3.1.116.1 $$\Z/2\Z$$ Not in database $6$ 6.0.1560896.1 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $8$ 8.2.24749176752.1 $$\Z/3\Z$$ Not in database $12$ 12.2.282621973446656.2 $$\Z/4\Z$$ Not in database

We only show fields where the torsion growth is primitive.