Properties

 Label 248430ec5 Conductor $248430$ Discriminant $1.789\times 10^{15}$ j-invariant $$\frac{524388516989299201}{3150}$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -139120973, 631580989106]) # or

sage: E = EllipticCurve("248430.ec1")

gp: E = ellinit([1, 0, 1, -139120973, 631580989106]) \\ or

gp: E = ellinit("248430.ec1")

magma: E := EllipticCurve([1, 0, 1, -139120973, 631580989106]); // or

magma: E := EllipticCurve("248430.ec1");

$$y^2+xy+y=x^3-139120973x+631580989106$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(6810, -3443\right)$$ $$\left(6920, 12342\right)$$ $$\hat{h}(P)$$ ≈ $2.1941560364658069960411151504$ $3.2868606281044484860760851373$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(\frac{27239}{4}, -\frac{27243}{8}\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(6810, -3368\right)$$, $$\left(6810, -3443\right)$$, $$\left(6920, 12342\right)$$, $$\left(6920, -19263\right)$$, $$\left(8880, 306097\right)$$, $$\left(8880, -314978\right)$$, $$\left(11922, 811069\right)$$, $$\left(11922, -822992\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$248430$$ = $$2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$1788788143929150$$ = $$2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{7} \cdot 13^{6}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{524388516989299201}{3150}$$ = $$2^{-1} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{-1} \cdot 431^{3} \cdot 1871^{3}$$ 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: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$7.2020294771326450637011896255$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.22910144231749578671779299490$$ 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: $$32$$  = $$1\cdot2\cdot2\cdot2\cdot2^{2}$$ 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: $$1$$ (rounded)

Modular invariants

Modular form 248430.2.a.ec

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);

$$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{8} + q^{9} - q^{10} - 4q^{11} + q^{12} + q^{15} + q^{16} - 2q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 28311552 $$\Gamma_0(N)$$-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);

$$L^{(2)}(E,1)/2!$$ ≈ $$13.199962726593672193400711030606833513$$

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
$$3$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$5$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$7$$ $$2$$ $$I_1^{*}$$ Additive -1 2 7 1
$$13$$ $$4$$ $$I_0^{*}$$ 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 X120.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^4\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 3 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 8 & 3 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 3 \end{array}\right)$ and has index 24.

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(3,20) if E.conductor().valuation(p)<2]

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

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4 and 8.
Its isogeny class 248430ec consists of 4 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 $K$ $2$ $$\Q(\sqrt{14})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{91})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{26})$$ $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{14}, \sqrt{26})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{15}, \sqrt{91})$$ $$\Z/8\Z$$ Not in database $4$ $$\Q(\sqrt{91}, \sqrt{210})$$ $$\Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.14093587427885056.14 $$\Z/8\Z$$ Not in database $8$ 8.8.227515262607360000.4 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/16\Z$$ Not in database $8$ Deg 8 $$\Z/6\Z$$ 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 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/16\Z$$ 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.