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This is a model for the modular curve $X_0(36)$.

## Minimal Weierstrass equation

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

gp: E = ellinit([0, 0, 0, 0, 1])

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

$$y^2=x^3+1$$ ## Mordell-Weil group structure

$\Z/{6}\Z$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(2, 3\right)$$ ## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-1, 0\right)$$, $$(0,\pm 1)$$, $$(2,\pm 3)$$ ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$36$$ = $2^{2} \cdot 3^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-432$ = $-1 \cdot 2^{4} \cdot 3^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$0$$ = $0$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z[(1+\sqrt{-3})/2]$$ (potential complex multiplication) Sato-Tate group: $N(\mathrm{U}(1))$ Faltings height: $-0.81541529607436205566864717947\dots$ Stable Faltings height: $-1.3211174284280379149898691959\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $4.2065463159763627835250572371\dots$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $6$  = $3\cdot2$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $6$ 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/factorial(ar)  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); Special value: $L(E,1)$ ≈ $0.70109105266272713058750953952426833642$

## Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - 4 q^{7} + 2 q^{13} + 8 q^{19} + O(q^{20})$$ sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 2 primes of bad reduction:

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$(j)_{-}$
$2$ $3$ $IV$ Additive -1 2 4 0
$3$ $2$ $III$ Additive 1 2 3 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
$2$ 2B 16.192.9.83
$3$ 3B.1.1 27.648.18.1

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

All $p$-adic regulators are identically $1$ since the rank is $0$.

## Iwasawa invariants

$p$ Reduction type 2 3 add add - - - -

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

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 $d=$ 2, 3 and 6.
Its isogeny class 36a consists of 4 curves linked by isogenies of degrees dividing 6.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-3})$$ $$\Z/2\Z \times \Z/6\Z$$ 2.0.3.1-144.1-CMa1 $4$ 4.2.1728.1 $$\Z/12\Z$$ Not in database $6$ 6.0.34992.1 $$\Z/6\Z \times \Z/6\Z$$ Not in database $8$ 8.0.2985984.1 $$\Z/4\Z \times \Z/12\Z$$ Not in database $9$ 9.3.918330048.1 $$\Z/18\Z$$ Not in database $12$ 12.0.84687918336.1 $$\Z/2\Z \times \Z/42\Z$$ Not in database $16$ 16.4.2337302235907620864.1 $$\Z/24\Z$$ Not in database $18$ 18.0.2529990231179046912.1 $$\Z/6\Z \times \Z/18\Z$$ Not in database

We only show fields where the torsion growth is primitive.