# Properties

 Label 1526.f2 Conductor 1526 Discriminant -58622816 j-invariant $$\frac{13806727199}{58622816}$$ CM no Rank 1 Torsion Structure $$\Z/{5}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 1, 50, 363]); // or
magma: E := EllipticCurve("1526e1");
sage: E = EllipticCurve([1, 1, 1, 50, 363]) # or
sage: E = EllipticCurve("1526e1")
gp: E = ellinit([1, 1, 1, 50, 363]) \\ or
gp: E = ellinit("1526e1")

$$y^2 + x y + y = x^{3} + x^{2} + 50 x + 363$$

## Mordell-Weil group structure

$$\Z\times \Z/{5}\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

 $$P$$ = $$\left(21, 95\right)$$ $$\hat{h}(P)$$ ≈ 2.46796364391

## Torsion generators

magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp: elltors(E)

$$\left(-3, 15\right)$$

## Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

$$\left(-3, 15\right)$$, $$\left(11, 43\right)$$, $$\left(21, 95\right)$$, $$\left(207, 2885\right)$$

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$1526$$ = $$2 \cdot 7 \cdot 109$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$-58622816$$ = $$-1 \cdot 2^{5} \cdot 7^{5} \cdot 109$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{13806727199}{58622816}$$ = $$2^{-5} \cdot 7^{-5} \cdot 109^{-1} \cdot 2399^{3}$$ 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: $$2.46796364391$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$1.41388278621$$ 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: $$25$$  = $$5\cdot5\cdot1$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] Torsion order: $$5$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form1526.2.a.f

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)

$$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{7} + q^{8} - 2q^{9} + q^{10} - 3q^{11} - q^{12} - 6q^{13} + q^{14} - q^{15} + q^{16} - 2q^{17} - 2q^{18} + O(q^{20})$$

#### Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
400 . This curve is $$\Gamma_0(N)$$-optimal.

#### 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])

$$L'(E,1)$$ ≈ $$3.48941131311$$

## 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$$(j)_{-}$$
$$2$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$7$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$109$$ $$1$$ $$I_{1}$$ Non-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 $$p$$ except those listed.

prime Image of Galois representation
$$5$$ B.1.1

## $p$-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 47 109 split ordinary ordinary split ordinary ordinary ordinary ss ordinary ss ordinary ordinary ordinary ordinary ordinary nonsplit 3 1 7 4 1 1 3 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 0 0 0 0

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 5.
Its isogeny class 1526.f consists of 2 curves linked by isogenies of degree 5.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{5}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.6104.1 $$\Z/10\Z$$ Not in database
6 6.0.227427812864.1 $$\Z/2\Z \times \Z/10\Z$$ 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.