# Properties

 Label 231280.g1 Conductor $231280$ Discriminant $-719966224262994329600$ j-invariant $-\frac{581453267835321}{73208248217600}$ CM no Rank $1$ Torsion Structure $\mathrm{Trivial}$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 0, 0, -372547, -1293927614]); // or
magma: E := EllipticCurve("231280g1");
sage: E = EllipticCurve([0, 0, 0, -372547, -1293927614]) # or
sage: E = EllipticCurve("231280g1")
gp: E = ellinit([0, 0, 0, -372547, -1293927614]) \\ or
gp: E = ellinit("231280g1")

$y^2 = x^{3} - 372547 x - 1293927614$

## Mordell-Weil group structure

$\Z$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

 $P$ = $\left(1505, 39424\right)$ $\hat{h}(P)$ ≈ 2.96979343254

## Integral points

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

$\left(1505, 39424\right)$

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] $N$ = $231280$ = $2^{4} \cdot 5 \cdot 7^{2} \cdot 59$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc $\Delta$ = $-719966224262994329600$ = $-1 \cdot 2^{24} \cdot 5^{2} \cdot 7^{4} \cdot 59^{5}$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j $j$ = $-\frac{581453267835321}{73208248217600}$ = $-1 \cdot 2^{-12} \cdot 3^{3} \cdot 5^{-2} \cdot 7^{2} \cdot 59^{-5} \cdot 7603^{3}$ $\text{End} (E)$ = $\Z$ (no Complex Multiplication) $\text{ST} (E)$ = $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E); sage: E.rank() $r$ = $1$ magma: Regulator(E); sage: E.regulator() $\text{Reg}$ ≈ $2.96979343254$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] $\Omega$ ≈ $0.0712275730201$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] $\prod_p c_p$ = $24$  = $2^{2}\cdot2\cdot3\cdot1$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] $\#E_{\text{tor}}$ = $1$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Ш$_{\text{an}}$ = $1$ (exact)

## Modular invariants

### Modular form 231280.2.1.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)

$q - 3q^{3} + q^{5} + 6q^{9} + 2q^{11} - 6q^{13} - 3q^{15} - q^{17} + q^{19} + O(q^{20})$

### Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
14653440 : curve is $\Gamma_0(N)$-optimal

### Special L-value attached to the curve

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])

$L'(E,1)$ ≈ $5.07674828569$

## 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$(j)_{-}$
$2$ $4$ $I_16^{*}$ Additive -1 4 24 12
$5$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$7$ $3$ $IV$ Additive 1 2 4 0
$59$ $1$ $I_{5}$ Non-split multiplicative 1 1 5 5

## 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]

No $p$-adic data exists for this curve.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has no rational isogenies. Its isogeny class 231280.g 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 3.1.2891.3 $\Z/2\Z$ Not in database
6 6.0.493114979.2 $\Z/2\Z \times \Z/2\Z$ 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.