Properties

 Label 35490bm7 Conductor $35490$ Discriminant $-6.548\times 10^{19}$ j-invariant $$\frac{42841933504271}{13565917968750}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, 123197, -388959244]) # or

sage: E = EllipticCurve("35490.bo8")

gp: E = ellinit([1, 0, 1, 123197, -388959244]) \\ or

gp: E = ellinit("35490.bo8")

magma: E := EllipticCurve([1, 0, 1, 123197, -388959244]); // or

magma: E := EllipticCurve("35490.bo8");

$$y^2+xy+y=x^3+123197x-388959244$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(1080, 31147\right)$$ $$\hat{h}(P)$$ ≈ $1.1904958346130465784281437391$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(\frac{2695}{4}, -\frac{2699}{8}\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(1080, 31147\right)$$, $$\left(1080, -32228\right)$$, $$\left(4096, 260324\right)$$, $$\left(4096, -264421\right)$$, $$\left(4330, 283022\right)$$, $$\left(4330, -287353\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$35490$$ = $$2 \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-65480094944824218750$$ = $$-1 \cdot 2 \cdot 3^{4} \cdot 5^{12} \cdot 7^{3} \cdot 13^{6}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{42841933504271}{13565917968750}$$ = $$2^{-1} \cdot 3^{-4} \cdot 5^{-12} \cdot 7^{-3} \cdot 11^{3} \cdot 3181^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1.1904958346130465784281437391$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.092007421717566439148361893287$$ 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: $$192$$  = $$1\cdot2^{2}\cdot( 2^{2} \cdot 3 )\cdot1\cdot2^{2}$$ 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 35490.2.a.bo

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);

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

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1327104 $$\Gamma_0(N)$$-optimal: no 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);

$$L'(E,1)$$ ≈ $$5.2576537107959426639178591094428158188$$

Local data

This elliptic curve is semistable. There are 5 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$$(j)_{-}$$
$$2$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$3$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4
$$5$$ $$12$$ $$I_{12}$$ Split multiplicative -1 1 12 12
$$7$$ $$1$$ $$I_{3}$$ Non-split multiplicative 1 1 3 3
$$13$$ $$4$$ $$I_0^{*}$$ 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 X13.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$ and has index 6.

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
$$3$$ B

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

$$p$$-adic regulators are not yet computed for curves that are not $$\Gamma_0$$-optimal.

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 nonsplit split split nonsplit ss add ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary 4 4 4 1 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, 3, 4, 6 and 12.
Its isogeny class 35490bm consists of 6 curves linked by isogenies of degrees dividing 12.

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 $K$ $2$ $$\Q(\sqrt{-14})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{91})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-26})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-39})$$ $$\Z/6\Z$$ Not in database $4$ $$\Q(\sqrt{-14}, \sqrt{-26})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-14}, \sqrt{-39})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database $4$ $$\Q(\sqrt{-21}, \sqrt{-39})$$ $$\Z/12\Z$$ Not in database $4$ $$\Q(\sqrt{6}, \sqrt{-26})$$ $$\Z/12\Z$$ Not in database $6$ 6.2.2075690448.9 $$\Z/6\Z$$ Not in database $8$ 8.0.14093587427885056.27 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/8\Z$$ Not in database $8$ 8.0.364024420171776.80 $$\Z/2\Z \times \Z/12\Z$$ Not in database $12$ Deg 12 $$\Z/3\Z \times \Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/12\Z$$ Not in database $12$ Deg 12 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/4\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/24\Z$$ Not in database $16$ Deg 16 $$\Z/24\Z$$ Not in database $18$ 18.0.5666378286557876363665906723550526937500000000.1 $$\Z/18\Z$$ 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.