# Properties

 Label 574.g2 Conductor $574$ Discriminant $7.081\times 10^{13}$ j-invariant $$\frac{801581275315909089}{70810888830976}$$ CM no Rank $1$ Torsion structure $$\Z/{7}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

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

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

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

$$y^2+xy+y=x^3-x^2-19353x+958713$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(61, 18\right)$$ $\hat{h}(P)$ ≈ $1.0673824816878454875840596007$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

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

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-121, 1292\right)$$, $$\left(-121, -1172\right)$$, $$\left(-37, 1292\right)$$, $$\left(-37, -1256\right)$$, $$\left(-9, 1068\right)$$, $$\left(-9, -1060\right)$$, $$\left(39, 492\right)$$, $$\left(39, -532\right)$$, $$\left(61, 18\right)$$, $$\left(61, -80\right)$$, $$\left(103, 172\right)$$, $$\left(103, -276\right)$$, $$\left(133, 784\right)$$, $$\left(133, -918\right)$$, $$\left(159, 1292\right)$$, $$\left(159, -1452\right)$$, $$\left(551, 12268\right)$$, $$\left(551, -12820\right)$$, $$\left(7019, 584392\right)$$, $$\left(7019, -591412\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$574$$ = $2 \cdot 7 \cdot 41$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $70810888830976$ = $2^{21} \cdot 7^{7} \cdot 41$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); 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: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.3977856380968835507713534157\dots$ Stable Faltings height: $1.3977856380968835507713534157\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $1.0673824816878454875840596007\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.60020819121169191031840301132\dots$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $147$  = $( 3 \cdot 7 )\cdot7\cdot1$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $7$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (exact) sage: r = E.rank(); sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: ar = ellanalyticrank(E); gp: ar[2]/factorial(ar[1])  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); Special value: $L'(E,1)$ ≈ $1.9219551259947258101377176144850767847$

## Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

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

magma: ModularForm(E);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 3528 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is semistable. There are 3 primes of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)[5]

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

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

sage: rho = E.galois_representation();

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

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

The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$7$ 7B.1.1 7.48.0.1

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

Note: $p$-adic regulator data only exists for primes $p\ge 5$ 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 less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{7}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 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 $8$ 8.2.14838034276107.1 $$\Z/21\Z$$ Not in database $12$ Deg 12 $$\Z/28\Z$$ Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.