# Properties

 Label 270802d1 Conductor $270802$ Discriminant $-2136356978$ j-invariant $$\frac{1297427375}{2540258}$$ CM no Rank $2$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, 215, -1777])

gp: E = ellinit([1, 1, 0, 215, -1777])

magma: E := EllipticCurve([1, 1, 0, 215, -1777]);

$$y^2+xy=x^3+x^2+215x-1777$$

## Mordell-Weil group structure

$\Z^2$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(53, 376\right)$$ $$\left(7, 8\right)$$ $\hat{h}(P)$ ≈ $1.1028084394998745284993997016$ $1.3952216492841157495513709149$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(7, 8\right)$$, $$\left(7, -15\right)$$, $$\left(11, 40\right)$$, $$\left(11, -51\right)$$, $$\left(53, 376\right)$$, $$\left(53, -429\right)$$, $$\left(257, 4008\right)$$, $$\left(257, -4265\right)$$, $$\left(28571, 4815165\right)$$, $$\left(28571, -4843736\right)$$

## 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: $-2136356978$ = $-1 \cdot 2 \cdot 7^{4} \cdot 23^{2} \cdot 29^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{1297427375}{2540258}$$ = $2^{-1} \cdot 5^{3} \cdot 7^{-4} \cdot 23^{-2} \cdot 29 \cdot 71^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.47472458991726400926476535196\dots$ Stable Faltings height: $-0.086491381747148328599113320100\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $2$ sage: E.regulator()  magma: Regulator(E); Regulator: $1.3623321127903556980333714437\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.76515419428834125165684377805\dots$ 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: $8$  = $1\cdot2^{2}\cdot2\cdot1$ 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: $1$ (rounded) 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: $L^{(2)}(E,1)/2!$ ≈ $8.3391530409219060134105227737427435816$

## Modular invariants

Modular form 270802.2.a.d

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 101760 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$7$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$23$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$29$ $1$ $II$ Additive 1 2 2 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 $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2G 8.2.0.1

## $p$-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 270802d 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 $K$ $3$ 3.1.6728.1 $$\Z/2\Z$$ Not in database $6$ 6.0.362127872.1 $$\Z/2\Z \times \Z/2\Z$$ 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.