Properties

 Label 102550q1 Conductor 102550 Discriminant 16408000 j-invariant $$\frac{1039509197}{131264}$$ CM no Rank 2 Torsion Structure $$\mathrm{Trivial}$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 0, -105, 325]); // or

magma: E := EllipticCurve("102550q1");

sage: E = EllipticCurve([1, 1, 0, -105, 325]) # or

sage: E = EllipticCurve("102550q1")

gp: E = ellinit([1, 1, 0, -105, 325]) \\ or

gp: E = ellinit("102550q1")

$$y^2 + x y = x^{3} + x^{2} - 105 x + 325$$

Mordell-Weil group structure

$$\Z^2$$

Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-6, 31\right)$$ $$\left(-5, 30\right)$$ $$\hat{h}(P)$$ ≈ 1.59770128861 1.78857936039

Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-6, 31\right)$$, $$\left(-6, -25\right)$$, $$\left(-5, 30\right)$$, $$\left(-5, -25\right)$$, $$\left(9, 10\right)$$, $$\left(9, -19\right)$$, $$\left(10, 15\right)$$, $$\left(10, -25\right)$$, $$\left(1170, 39455\right)$$, $$\left(1170, -40625\right)$$, $$\left(3090, 170255\right)$$, $$\left(3090, -173345\right)$$

Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$102550$$ = $$2 \cdot 5^{2} \cdot 7 \cdot 293$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$16408000$$ = $$2^{6} \cdot 5^{3} \cdot 7 \cdot 293$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{1039509197}{131264}$$ = $$2^{-6} \cdot 7^{-1} \cdot 293^{-1} \cdot 1013^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$2$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$0.889192338481$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$2.1213807632$$ 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: $$4$$  = $$2\cdot2\cdot1\cdot1$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$1$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (rounded)

Modular invariants

Modular form 102550.2.a.h

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)

$$q - q^{2} - q^{3} + q^{4} + q^{6} + q^{7} - q^{8} - 2q^{9} - q^{12} + 2q^{13} - q^{14} + q^{16} - 3q^{17} + 2q^{18} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 20736 $$\Gamma_0(N)$$-optimal: yes 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])

$$L^{(2)}(E,1)/2!$$ ≈ $$7.54526208654$$

Local data

This elliptic curve is not semistable.

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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{6}$$ Non-split multiplicative 1 1 6 6
$$5$$ $$2$$ $$III$$ Additive -1 2 3 0
$$7$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$293$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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 $$p$$ .

$p$-adic data

$p$-adic regulators

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

Note: $$p$$-adic regulator data only exists for primes $$p\ge5$$ 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 47 293 nonsplit ordinary add split ss ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary split 5 2 - 3 2,2 2 2 2,2 2 2 2 2 2 2 2 3 0 0 - 0 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 no rational isogenies. Its isogeny class 102550q consists of this curve only.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.3.41020.1 $$\Z/2\Z$$ Not in database
6 6.6.69021909208000.1 $$\Z/2\Z \times \Z/2\Z$$ 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.