# Properties

 Label 270802l1 Conductor $270802$ Discriminant $-9.193\times 10^{22}$ j-invariant $$\frac{5805798253576046271027}{3769183309498679296}$$ CM no Rank $1$ Torsion structure trivial

# Learn more

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, 10858689, 4804578279])

gp: E = ellinit([1, -1, 1, 10858689, 4804578279])

magma: E := EllipticCurve([1, -1, 1, 10858689, 4804578279]);

$$y^2+xy+y=x^3-x^2+10858689x+4804578279$$

## Mordell-Weil group structure

$\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(2771, 235606\right)$$ $\hat{h}(P)$ ≈ $0.49935918817067736161828035283$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-429, 8406\right)$$, $$\left(-429, -7978\right)$$, $$\left(2771, 235606\right)$$, $$\left(2771, -238378\right)$$, $$\left(485971, 338542806\right)$$, $$\left(485971, -339028778\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$270802$$ = $2 \cdot 7 \cdot 23 \cdot 29^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-91926611735363289350144$ = $-1 \cdot 2^{38} \cdot 7^{2} \cdot 23^{4} \cdot 29^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{5805798253576046271027}{3769183309498679296}$$ = $2^{-38} \cdot 3^{3} \cdot 7^{-2} \cdot 23^{-4} \cdot 569^{3} \cdot 10529^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $3.0951572542575033427910590248\dots$ Stable Faltings height: $2.2533332967608848359952410167\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.49935918817067736161828035283\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.066935886072269563573150360105\dots$ 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: $304$  = $( 2 \cdot 19 )\cdot2\cdot2\cdot2$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $1$ 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: $L'(E,1)$ ≈ $10.161215117474177620681451062111936336$

## Modular invariants

Modular form 270802.2.a.l

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);

$$q + q^{2} - 3q^{3} + q^{4} + 3q^{5} - 3q^{6} - q^{7} + q^{8} + 6q^{9} + 3q^{10} + 5q^{11} - 3q^{12} + q^{13} - q^{14} - 9q^{15} + q^{16} + 2q^{17} + 6q^{18} - 4q^{19} + O(q^{20})$$

For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 63124992 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 4 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$(j)_{-}$
$2$ $38$ $I_{38}$ Split multiplicative -1 1 38 38
$7$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$23$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$29$ $2$ $III$ Additive -1 2 3 0

## 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 $\GL(2,\Z_\ell)$ for all primes $\ell$.

## $p$-adic regulators

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

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has no rational isogenies. Its isogeny class 270802l consists of this curve only.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.116.1 $$\Z/2\Z$$ Not in database $6$ 6.0.1560896.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ Deg 8 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ 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.