# Properties

 Label 102550bd1 Conductor 102550 Discriminant -22432812500 j-invariant $$-\frac{5151505}{57428}$$ CM no Rank 1 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 1, -263, 7281]); // or

magma: E := EllipticCurve("102550bd1");

sage: E = EllipticCurve([1, 1, 1, -263, 7281]) # or

sage: E = EllipticCurve("102550bd1")

gp: E = ellinit([1, 1, 1, -263, 7281]) \\ or

gp: E = ellinit("102550bd1")

$$y^2 + x y + y = x^{3} + x^{2} - 263 x + 7281$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(19, 88\right)$$ $$\hat{h}(P)$$ ≈ 2.21896289621

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(19, 88\right)$$, $$\left(19, -108\right)$$

## 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: $$-22432812500$$ = $$-1 \cdot 2^{2} \cdot 5^{8} \cdot 7^{2} \cdot 293$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{5151505}{57428}$$ = $$-1 \cdot 2^{-2} \cdot 5 \cdot 7^{-2} \cdot 101^{3} \cdot 293^{-1}$$ 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: $$2.21896289621$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$1.02534245838$$ 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: $$4$$  = $$2\cdot1\cdot2\cdot1$$ 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: $$1$$ (exact)

## Modular invariants

#### Modular form 102550.2.a.s

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)

$$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{7} + q^{8} - 2q^{9} - q^{11} - q^{12} + 4q^{13} + q^{14} + q^{16} - 2q^{17} - 2q^{18} + q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 66240 $$\Gamma_0(N)$$-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])

$$L'(E,1)$$ ≈ $$9.10078748419$$

## 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$5$$ $$1$$ $$IV^{*}$$ Additive -1 2 8 0
$$7$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$293$$ $$1$$ $$I_{1}$$ 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 split ordinary add split ordinary ordinary ordinary ordinary ss ss ordinary ordinary ordinary ordinary ordinary split 3 3 - 2 1 3 1 1 1,1 1,1 3 1 1 1 1 2 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 102550bd 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 3.1.29300.3 $$\Z/2\Z$$ Not in database
6 6.0.1006150280000.3 $$\Z/2\Z \times \Z/2\Z$$ 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.