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This is a model for the modular curve $X_0(17)$.

## Simplified equation

 $$y^2+xy+y=x^3-x^2-x-14$$ y^2+xy+y=x^3-x^2-x-14 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-x^2z-xz^2-14z^3$$ y^2z+xyz+yz^2=x^3-x^2z-xz^2-14z^3 (dehomogenize, simplify) $$y^2=x^3-11x-890$$ y^2=x^3-11x-890 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, -1, 1, -1, -14])

gp: E = ellinit([1, -1, 1, -1, -14])

magma: E := EllipticCurve([1, -1, 1, -1, -14]);

oscar: E = EllipticCurve([1, -1, 1, -1, -14])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z/{4}\Z$$

magma: MordellWeilGroup(E);

## Torsion generators

$$\left(7, 13\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(7, 13\right)$$, $$\left(7, -21\right)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$17$$ = $17$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E);  oscar: conductor(E) Discriminant: $-83521$ = $-1 \cdot 17^{4}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$-\frac{35937}{83521}$$ = $-1 \cdot 3^{3} \cdot 11^{3} \cdot 17^{-4}$ 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: $-0.37663612094134348095743300423\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-0.37663612094134348095743300423\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E)

## BSD invariants

 Analytic rank: $0$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $1$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $1.5470797535511201732095790050\dots$ 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: $4$  = $2^{2}$ 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: $4$ 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: $L(E,1)$ ≈ $0.38676993838778004330239475124$ 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

$\displaystyle 0.386769938 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.547080 \cdot 1.000000 \cdot 4}{4^2} \approx 0.386769938$

# 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

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

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: 1
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 is only one prime of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$17$ $4$ $I_{4}$ Split multiplicative -1 1 4 4

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 16.96.0.32

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 = [[191, 478, 974, 279], [1, 64, 0, 1], [174, 863, 739, 498], [9, 124, 756, 745], [513, 12, 0, 1], [1025, 64, 1024, 65], [1, 0, 64, 1], [57, 16, 176, 393]]

GL(2,Integers(1088)).subgroup(gens)

Gens := [[191, 478, 974, 279], [1, 64, 0, 1], [174, 863, 739, 498], [9, 124, 756, 745], [513, 12, 0, 1], [1025, 64, 1024, 65], [1, 0, 64, 1], [57, 16, 176, 393]];

sub<GL(2,Integers(1088))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$1088 = 2^{6} \cdot 17$$, index $1536$, genus $53$, and generators

$\left(\begin{array}{rr} 191 & 478 \\ 974 & 279 \end{array}\right),\left(\begin{array}{rr} 1 & 64 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 174 & 863 \\ 739 & 498 \end{array}\right),\left(\begin{array}{rr} 9 & 124 \\ 756 & 745 \end{array}\right),\left(\begin{array}{rr} 513 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1025 & 64 \\ 1024 & 65 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 64 & 1 \end{array}\right),\left(\begin{array}{rr} 57 & 16 \\ 176 & 393 \end{array}\right)$.

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

The torsion field $K:=\Q(E)$ is a degree-$320864256$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1088\Z)$.

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 17a consists of 4 curves linked by isogenies of degrees dividing 4.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-1})$$ $$\Z/2\Z \oplus \Z/4\Z$$ 2.0.4.1-289.2-a1 $4$ $$\Q(\zeta_{8})$$ $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $4$ 4.2.1156.1 $$\Z/8\Z$$ Not in database $8$ 8.0.5473632256.2 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.0.21381376.2 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.2.182660427.1 $$\Z/12\Z$$ Not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $16$ 16.0.29960650073923649536.2 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/16\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/12\Z$$ 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 2 17 ord split 0 1 2 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

## $p$-adic regulators

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

This is the only elliptic curve over $\Q$ of prime conductor $N$ and discriminant $\Delta = \pm N^4$, and one of only four prime-conductor curves of discriminant $\pm N^e$ with $e > 2$: the others are the modular curves $X_0(N)$ for N=11 [11.a2] and N=19 [19.a2], and the curve [37.b2] (with $e=5,3,3$ respectively -- in each case, as here with $(N,e)=(17,4)$, the exponent $e$ is the numerator of $(N-1)/12$).