# Properties

 Label 17a3 Conductor $17$ Discriminant $17$ j-invariant $$\frac{82483294977}{17}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy+y=x^3-x^2-91x-310$$ y^2+xy+y=x^3-x^2-91x-310 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-x^2z-91xz^2-310z^3$$ y^2z+xyz+yz^2=x^3-x^2z-91xz^2-310z^3 (dehomogenize, simplify) $$y^2=x^3-1451x-21274$$ y^2=x^3-1451x-21274 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, -1, 1, -91, -310])

gp: E = ellinit([1, -1, 1, -91, -310])

magma: E := EllipticCurve([1, -1, 1, -91, -310]);

oscar: E = elliptic_curve([1, -1, 1, -91, -310])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

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

magma: MordellWeilGroup(E);

## Torsion generators

$$\left(-\frac{21}{4}, \frac{17}{8}\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

None

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$17$$ = $17$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $17$ = $17$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{82483294977}{17}$$ = $3^{3} \cdot 17^{-1} \cdot 1451^{3}$ 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) $abc$ quality: $1.0313050753900588\dots$ Szpiro ratio: $8.871856289039895\dots$

## 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[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: $1$ ( 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

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

# 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: 4
comment: Modular degree

sage: E.modular_degree()

gp: ellmoddegree(E)

magma: ModularDegree(E);

$\Gamma_0(N)$-optimal: no
Manin constant: 2
comment: Manin constant

magma: ManinConstant(E);

## Local data

This elliptic curve is semistable. There is only one prime $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $v_p(N)$ $v_p(\Delta)$ $v_p(\mathrm{den}(j))$
$17$ $1$ $I_{1}$ 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 32.96.0.18

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 = [[378, 63, 561, 328], [1, 64, 0, 1], [511, 64, 714, 1], [15, 286, 314, 547], [1025, 64, 1024, 65], [1, 0, 64, 1], [59, 10, 574, 503], [807, 210, 898, 1071]]

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

Gens := [[378, 63, 561, 328], [1, 64, 0, 1], [511, 64, 714, 1], [15, 286, 314, 547], [1025, 64, 1024, 65], [1, 0, 64, 1], [59, 10, 574, 503], [807, 210, 898, 1071]];

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

$\left(\begin{array}{rr} 378 & 63 \\ 561 & 328 \end{array}\right),\left(\begin{array}{rr} 1 & 64 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 511 & 64 \\ 714 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 286 \\ 314 & 547 \end{array}\right),\left(\begin{array}{rr} 1025 & 64 \\ 1024 & 65 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 64 & 1 \end{array}\right),\left(\begin{array}{rr} 59 & 10 \\ 574 & 503 \end{array}\right),\left(\begin{array}{rr} 807 & 210 \\ 898 & 1071 \end{array}\right)$.

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

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

The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.

$\ell$ Reduction type Serre weight Serre conductor
$17$ split multiplicative $18$ $$1$$

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

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{17})$$ $$\Z/2\Z \oplus \Z/2\Z$$ 2.2.17.1-17.1-a7 $2$ $$\Q(\sqrt{-1})$$ $$\Z/4\Z$$ 2.0.4.1-289.2-a6 $2$ $$\Q(\sqrt{-17})$$ $$\Z/4\Z$$ not in database $4$ 4.2.19652.1 $$\Z/2\Z \oplus \Z/4\Z$$ not in database $4$ $$\Q(i, \sqrt{17})$$ $$\Z/2\Z \oplus \Z/4\Z$$ not in database $4$ 4.0.4352.2 $$\Z/8\Z$$ not in database $8$ 8.0.6179217664.3 $$\Z/4\Z \oplus \Z/4\Z$$ not in database $8$ 8.0.1581879721984.3 $$\Z/2\Z \oplus \Z/8\Z$$ not in database $8$ 8.0.5473632256.2 $$\Z/2\Z \oplus \Z/8\Z$$ not in database $8$ 8.2.182660427.1 $$\Z/6\Z$$ not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/8\Z$$ not in database $16$ deg 16 $$\Z/16\Z$$ not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ not in database $16$ deg 16 $$\Z/12\Z$$ not in database $16$ deg 16 $$\Z/12\Z$$ 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 $0$.