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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)  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)

## 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  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[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E);  oscar: tamagawa_numbers(E) Torsion order: $2$ comment: Torsion order  sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E));  oscar: prod(torsion_structure(E)) Analytic order of Ш: $1$ (exact) 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("anayltic 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);

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)

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

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

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

## $p$-adic regulators

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 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

## 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.

## 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.

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 $E$.