# Properties

 Label 574.g2 Conductor 574 Discriminant 70810888830976 j-invariant $$\frac{801581275315909089}{70810888830976}$$ CM no Rank 1 Torsion Structure $$\Z/{7}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, -19353, 958713]); // or

magma: E := EllipticCurve("574i1");

sage: E = EllipticCurve([1, -1, 1, -19353, 958713]) # or

sage: E = EllipticCurve("574i1")

gp: E = ellinit([1, -1, 1, -19353, 958713]) \\ or

gp: E = ellinit("574i1")

$$y^2 + x y + y = x^{3} - x^{2} - 19353 x + 958713$$

## Mordell-Weil group structure

$$\Z\times \Z/{7}\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-121, 1292\right)$$ $$\hat{h}(P)$$ ≈ 1.06738248169

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(103, 172\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-121, 1292\right)$$, $$\left(-37, 1292\right)$$, $$\left(-9, 1068\right)$$, $$\left(39, 492\right)$$, $$\left(61, 18\right)$$, $$\left(103, 172\right)$$, $$\left(133, 784\right)$$, $$\left(159, 1292\right)$$, $$\left(551, 12268\right)$$, $$\left(7019, 584392\right)$$

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

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$574$$ = $$2 \cdot 7 \cdot 41$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$70810888830976$$ = $$2^{21} \cdot 7^{7} \cdot 41$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{801581275315909089}{70810888830976}$$ = $$2^{-21} \cdot 3^{3} \cdot 7^{-7} \cdot 19^{3} \cdot 41^{-1} \cdot 43^{3} \cdot 379^{3}$$ 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: $$1.06738248169$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.600208191212$$ 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: $$147$$  = $$( 3 \cdot 7 )\cdot7\cdot1$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$7$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form574.2.a.g

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)

$$q + q^{2} - 3q^{3} + q^{4} - q^{5} - 3q^{6} + q^{7} + q^{8} + 6q^{9} - q^{10} - 2q^{11} - 3q^{12} + q^{14} + 3q^{15} + q^{16} - 3q^{17} + 6q^{18} - 8q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 3528 $$\Gamma_0(N)$$-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])

$$L'(E,1)$$ ≈ $$1.92195512599$$

## 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$$(j)_{-}$$
$$2$$ $$21$$ $$I_{21}$$ Split multiplicative -1 1 21 21
$$7$$ $$7$$ $$I_{7}$$ Split multiplicative -1 1 7 7
$$41$$ $$1$$ $$I_{1}$$ 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
$$7$$ 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 split ss ordinary split ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary nonsplit ordinary ordinary 2 1,1 1 2 1 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

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 7.
Its isogeny class 574.g consists of 2 curves linked by isogenies of degree 7.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{7}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.3.2296.1 $$\Z/14\Z$$ Not in database
6 6.6.12103630336.1 $$\Z/2\Z \times \Z/14\Z$$ 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.