# Properties

 Label 65.a1 Conductor $65$ Discriminant $65$ j-invariant $$\frac{117649}{65}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

This is a model for the quotient of the modular curve $X_0(65)$ by its Fricke involution $w_{65}$; this quotient is also denoted $X_0^+(65)$.

## Simplified equation

 $$y^2+xy=x^3-x$$ y^2+xy=x^3-x (homogenize, simplify) $$y^2z+xyz=x^3-xz^2$$ y^2z+xyz=x^3-xz^2 (dehomogenize, simplify) $$y^2=x^3-1323x+3942$$ y^2=x^3-1323x+3942 (homogenize, minimize)

comment: Define the curve

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

gp: E = ellinit([1, 0, 0, -1, 0])

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

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

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

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

magma: MordellWeilGroup(E);

### Infinite order Mordell-Weil generator and height

 $P$ = $$\left(1, 0\right)$$ (1, 0) $\hat{h}(P)$ ≈ $0.37551409866126632180447287682$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Torsion generators

$$\left(0, 0\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(-1, 1\right)$$, $$\left(-1, 0\right)$$, $$\left(0, 0\right)$$, $$\left(1, 0\right)$$, $$\left(1, -1\right)$$, $$\left(4, 6\right)$$, $$\left(4, -10\right)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$65$$ = $5 \cdot 13$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $65$ = $5 \cdot 13$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{117649}{65}$$ = $5^{-1} \cdot 7^{6} \cdot 13^{-1}$ comment: j-invariant  sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E);  oscar: j_invariant(E) Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.96161523729216172163187933912\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-0.96161523729216172163187933912\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $0.9568076435408001\dots$ Szpiro ratio: $2.7969280614023564\dots$

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $0.37551409866126632180447287682\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $5.3828534705718009941152235294\dots$ comment: Real Period  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); Tamagawa product: $1$ comment: Tamagawa numbers  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E);  oscar: tamagawa_numbers(E) Torsion order: $2$ comment: Torsion order  sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E));  oscar: prod(torsion_structure(E)[1]) Analytic order of Ш: $1$ ( rounded) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L'(E,1)$ ≈ $0.50533434230685977745953442461$ comment: Special L-value  r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: [r,L1r] = ellanalyticrank(E); L1r/r!  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

## BSD formula

$\displaystyle 0.505334342 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 5.382853 \cdot 0.375514 \cdot 1}{2^2} \approx 0.505334342$

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)

E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;

Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();

omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();

assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))

/* self-contained Magma code snippet for the BSD formula (checks rank, computes analyiic sha) */

E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;

sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);

reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);

assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);

## Modular invariants

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

comment: q-expansion of modular form

sage: E.q_eigenform(20)

\\ actual modular form, use for small N

[mf,F] = mffromell(E)

Ser(mfcoefs(mf,20),q)

\\ or just the series

Ser(ellan(E,20),q)*q

magma: ModularForm(E);

For more coefficients, see the Downloads section to the right.

Modular degree: 2
comment: Modular degree

sage: E.modular_degree()

gp: ellmoddegree(E)

magma: ModularDegree(E);

$\Gamma_0(N)$-optimal: yes
Manin constant: 1
comment: Manin constant

magma: ManinConstant(E);

## Local data

This elliptic curve is semistable. There are 2 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$5$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$13$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

comment: Local data

sage: E.local_data()

gp: ellglobalred(E)[5]

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

oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]

## Galois representations

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 8.12.0.22

comment: mod p Galois image

sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]

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

gens = [[513, 8, 512, 9], [1, 0, 8, 1], [1, 8, 0, 1], [328, 3, 205, 2], [1, 4, 4, 17], [422, 1, 231, 4], [3, 8, 28, 75], [391, 8, 260, 1], [5, 8, 48, 77], [261, 8, 0, 1]]

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

Gens := [[513, 8, 512, 9], [1, 0, 8, 1], [1, 8, 0, 1], [328, 3, 205, 2], [1, 4, 4, 17], [422, 1, 231, 4], [3, 8, 28, 75], [391, 8, 260, 1], [5, 8, 48, 77], [261, 8, 0, 1]];

sub<GL(2,Integers(520))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$520 = 2^{3} \cdot 5 \cdot 13$$, index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 513 & 8 \\ 512 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 328 & 3 \\ 205 & 2 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 422 & 1 \\ 231 & 4 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 28 & 75 \end{array}\right),\left(\begin{array}{rr} 391 & 8 \\ 260 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 261 & 8 \\ 0 & 1 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

The torsion field $K:=\Q(E[520])$ is a degree-$402554880$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/520\Z)$.

## Isogenies

gp: ellisomat(E)

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.

## Twists

This elliptic curve is its own minimal quadratic twist.

## 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 $K$ $2$ $$\Q(\sqrt{65})$$ $$\Z/2\Z \oplus \Z/2\Z$$ 2.2.65.1-65.1-a2 $4$ 4.0.1040.2 $$\Z/4\Z$$ Not in database $8$ 8.4.1206702250000.5 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.4569760000.3 $$\Z/2\Z \oplus \Z/4\Z$$ 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.

## 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

## $p$-adic regulators

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

This curve $E$, together with the $2$-isogenous curve [65.a2], have the minimal conductor for elliptic curves over $\bf Q$ whose Mordell-Weil group have positive rank and nontrivial torsion. This is related to the identification of $E$ with the quotient of the modular curve $X_0(65)$ by its Fricke involution $w_{65}$: the Atkin-Lehner involutions $w_5$ and $w_{13}$ descend to an involution of $E$ that has no fixed points (because each of $5$ and $13$ is a quadratic nonresidue of the other) and thus must be translation by a rational $2$-torsion point on $E$.