Properties

 Label 212160.cc5 Conductor 212160 Discriminant 3737761374524767469568000000 j-invariant $$\frac{1897660325010178513043539489}{14258428094958372000000}$$ CM no Rank 1 Torsion Structure $$\Z/{2}\Z$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([0, -1, 0, -1650738305, -25645921710975]); // or

magma: E := EllipticCurve("212160gf3");

sage: E = EllipticCurve([0, -1, 0, -1650738305, -25645921710975]) # or

sage: E = EllipticCurve("212160gf3")

gp: E = ellinit([0, -1, 0, -1650738305, -25645921710975]) \\ or

gp: E = ellinit("212160gf3")

$$y^2 = x^{3} - x^{2} - 1650738305 x - 25645921710975$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-\frac{41863264}{1681}, -\frac{9195071145}{68921}\right)$$ $$\hat{h}(P)$$ ≈ 8.79204070366

Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(46881, 0\right)$$

Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(46881, 0\right)$$

Note: only one of each pair $\pm P$ is listed.

Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$212160$$ = $$2^{6} \cdot 3 \cdot 5 \cdot 13 \cdot 17$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$3737761374524767469568000000$$ = $$2^{26} \cdot 3^{2} \cdot 5^{6} \cdot 13^{12} \cdot 17$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{1897660325010178513043539489}{14258428094958372000000}$$ = $$2^{-8} \cdot 3^{-2} \cdot 5^{-6} \cdot 11^{6} \cdot 13^{-12} \cdot 17^{-1} \cdot 23^{3} \cdot 444863^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$1$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$8.79204070366$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.0236960918458$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$96$$  = $$2^{2}\cdot2\cdot( 2 \cdot 3 )\cdot2\cdot1$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$2$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

Modular invariants

Modular form 212160.2.a.cc

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

$$q - q^{3} + q^{5} - 4q^{7} + q^{9} - q^{13} - q^{15} + q^{17} + 4q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 159252480 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar[2]/factorial(ar[1])

$$L'(E,1)$$ ≈ $$5.00008809662$$

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

sage: E.local_data()

gp: ellglobalred(E)[5]

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$4$$ $$I_16^{*}$$ Additive 1 6 26 8
$$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$5$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$13$$ $$2$$ $$I_{12}$$ Non-split multiplicative 1 1 12 12
$$17$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1

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 X13e.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 5 & 5 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 3 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$ and has index 12.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

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.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 4, 6 and 12.
Its isogeny class 212160.cc consists of 8 curves linked by isogenies of degrees dividing 12.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{2})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{-6})$$ $$\Z/6\Z$$ Not in database
$$\Q(\sqrt{17})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
$$\Q(\sqrt{34})$$ $$\Z/4\Z$$ Not in database
4 $$\Q(\sqrt{2}, \sqrt{17})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{2}, \sqrt{-3})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{-6}, \sqrt{17})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
$$\Q(\sqrt{-6}, \sqrt{34})$$ $$\Z/12\Z$$ Not in database
6 6.2.10100390971392.4 $$\Z/12\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.