# Properties

 Label 65.a2 Conductor $65$ Discriminant $-4225$ j-invariant $$\frac{6967871}{4225}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

This is a model for the quotient of the modular curve $X_0(65)$ by its group $\langle w_5, w_{13} \rangle$ of Atkin-Lehner involutions.

## Simplified equation

 $$y^2+xy=x^3+4x+1$$ y^2+xy=x^3+4x+1 (homogenize, simplify) $$y^2z+xyz=x^3+4xz^2+z^3$$ y^2z+xyz=x^3+4xz^2+z^3 (dehomogenize, simplify) $$y^2=x^3+5157x+31158$$ y^2=x^3+5157x+31158 (homogenize, minimize)

sage: E = EllipticCurve([1, 0, 0, 4, 1])

gp: E = ellinit([1, 0, 0, 4, 1])

magma: E := EllipticCurve([1, 0, 0, 4, 1]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z \oplus \Z/{2}\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(1, 2\right)$$ (1, 2) $\hat{h}(P)$ ≈ $0.18775704933063316090223643841$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-\frac{1}{4}, \frac{1}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(0, 1\right)$$, $$\left(0, -1\right)$$, $$\left(1, 2\right)$$, $$\left(1, -3\right)$$, $$\left(3, 5\right)$$, $$\left(3, -8\right)$$, $$\left(11, 32\right)$$, $$\left(11, -43\right)$$, $$\left(16, 57\right)$$, $$\left(16, -73\right)$$, $$\left(393, 7597\right)$$, $$\left(393, -7990\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$65$$ = $5 \cdot 13$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-4225$ = $-1 \cdot 5^{2} \cdot 13^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{6967871}{4225}$$ = $5^{-2} \cdot 13^{-2} \cdot 191^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.61504164701218906692326327839\dots$ Stable Faltings height: $-0.61504164701218906692326327839\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.18775704933063316090223643841\dots$ sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Real period: $2.6914267352859004970576117647\dots$ 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: $4$  = $2\cdot2$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $2$ 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: $L'(E,1)$ ≈ $0.50533434230685977745953442461$

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 4 $\Gamma_0(N)$-optimal: no 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$(j)_{-}$
$5$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$13$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

## 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 8.12.0.37
sage: gens = [[391, 8, 327, 17], [1, 0, 8, 1], [5, 8, 48, 77], [513, 8, 512, 9], [3, 8, 10, 27], [261, 8, 2, 17], [41, 2, 0, 1], [417, 6, 0, 1], [1, 8, 0, 1]]

sage: GL(2,Integers(520)).subgroup(gens)

magma: Gens := [[391, 8, 327, 17], [1, 0, 8, 1], [5, 8, 48, 77], [513, 8, 512, 9], [3, 8, 10, 27], [261, 8, 2, 17], [41, 2, 0, 1], [417, 6, 0, 1], [1, 8, 0, 1]];

magma: sub<GL(2,Integers(520))|Gens>;

The image of the adelic Galois representation has level $520$, index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 391 & 8 \\ 327 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 513 & 8 \\ 512 & 9 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 10 & 27 \end{array}\right),\left(\begin{array}{rr} 261 & 8 \\ 2 & 17 \end{array}\right),\left(\begin{array}{rr} 41 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 417 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$

## $p$-adic regulators

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

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.

## 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 ord ord nonsplit ord ord nonsplit ord ord ord ord ord ord ord ord ord 2 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 65.a consists of 2 curves linked by isogenies of degree 2.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-1})$$ $$\Z/2\Z \oplus \Z/2\Z$$ 2.0.4.1-4225.5-a1 $4$ 4.2.16900.2 $$\Z/4\Z$$ Not in database $8$ 8.0.4569760000.3 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.276889600.3 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.2.39039316875.1 $$\Z/6\Z$$ Not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $16$ deg 16 $$\Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ 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.