Properties

 Label 17424.bu5 Conductor $17424$ Discriminant $-61990462512$ j-invariant $$\frac{2048}{3}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

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Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, 726, -9317])

gp: E = ellinit([0, 0, 0, 726, -9317])

magma: E := EllipticCurve([0, 0, 0, 726, -9317]);

$$y^2=x^3+726x-9317$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(\frac{60779}{289}, \frac{15099120}{4913}\right)$$ $$\hat{h}(P)$$ ≈ $9.9827470536769612721093327142$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(11, 0\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(11, 0\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$17424$$ = $$2^{4} \cdot 3^{2} \cdot 11^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-61990462512$$ = $$-1 \cdot 2^{4} \cdot 3^{7} \cdot 11^{6}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{2048}{3}$$ = $$2^{11} \cdot 3^{-1}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$0.75632790437341019036096766989\dots$$ Stable Faltings height: $$-1.2229749365464783638400374447\dots$$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$9.9827470536769612721093327142\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.58690294058452530681259666371\dots$$ 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: $$4$$  = $$1\cdot2\cdot2$$ 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)

Modular invariants

Modular form 17424.2.a.bu

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);

$$q + 2q^{5} + 2q^{13} + 2q^{17} - 4q^{19} + O(q^{20})$$

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 10240 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

Special L-value

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);

$$L'(E,1)$$ ≈ $$5.8589036009145146652501785346148848037$$

Local data

This elliptic curve is not semistable. There are 3 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$$(j)_{-}$$
$$2$$ $$1$$ $$II$$ Additive 1 4 4 0
$$3$$ $$2$$ $$I_1^{*}$$ Additive -1 2 7 1
$$11$$ $$2$$ $$I_0^{*}$$ Additive -1 2 6 0

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X228.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^4\Z_2)$ generated by $\left(\begin{array}{rr} 5 & 5 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 3 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 3 \end{array}\right)$ and has index 48.

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 mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$2$$ B

$p$-adic data

$p$-adic regulators

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

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 add add ordinary ss add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss - - 1 1,1 - 1 1 1 1 1 1 1 1 1 1,1 - - 0 0,0 - 0 0 0 0 0 0 0 0 0 0,0

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4 and 8.
Its isogeny class 17424.bu consists of 6 curves linked by isogenies of degrees dividing 8.

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 $K$ $2$ $$\Q(\sqrt{-3})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{11})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-33})$$ $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-3}, \sqrt{11})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(i, \sqrt{33})$$ $$\Z/8\Z$$ Not in database $4$ $$\Q(\sqrt{3}, \sqrt{-11})$$ $$\Z/8\Z$$ Not in database $8$ 8.0.2732361984.3 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.4.43717791744.4 $$\Z/8\Z$$ Not in database $8$ 8.0.303595776.1 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.2.10623423393792.1 $$\Z/6\Z$$ Not in database $16$ 16.0.1911245314971754561536.7 $$\Z/4\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ 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.