# Properties

 Label 350.d1 Conductor $350$ Discriminant $-62720000$ j-invariant $$-\frac{1026590625}{100352}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -180, 1047])

gp: E = ellinit([1, -1, 1, -180, 1047])

magma: E := EllipticCurve([1, -1, 1, -180, 1047]);

$$y^2+xy+y=x^3-x^2-180x+1047$$

## Mordell-Weil group structure

$\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(-1, 35\right)$$ $\hat{h}(P)$ ≈ $0.012846808401829569970207278286$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-15, 21\right)$$, $$\left(-15, -7\right)$$, $$\left(-11, 45\right)$$, $$\left(-11, -35\right)$$, $$\left(-1, 35\right)$$, $$\left(-1, -35\right)$$, $$\left(3, 21\right)$$, $$\left(3, -25\right)$$, $$\left(5, 13\right)$$, $$\left(5, -19\right)$$, $$\left(9, 5\right)$$, $$\left(9, -15\right)$$, $$\left(13, 21\right)$$, $$\left(13, -35\right)$$, $$\left(19, 55\right)$$, $$\left(19, -75\right)$$, $$\left(29, 125\right)$$, $$\left(29, -155\right)$$, $$\left(69, 525\right)$$, $$\left(69, -595\right)$$, $$\left(209, 2905\right)$$, $$\left(209, -3115\right)$$, $$\left(373, 7005\right)$$, $$\left(373, -7379\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$350$$ = $2 \cdot 5^{2} \cdot 7$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-62720000$ = $-1 \cdot 2^{11} \cdot 5^{4} \cdot 7^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{1026590625}{100352}$$ = $-1 \cdot 2^{-11} \cdot 3^{3} \cdot 5^{5} \cdot 7^{-2} \cdot 23^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.23710318408171966996539577908\dots$ Stable Faltings height: $-0.29937612006298045490152399866\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.012846808401829569970207278286\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $1.9197624983313408116306920193\dots$ 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: $66$  = $11\cdot3\cdot2$ 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: $L'(E,1)$ ≈ $1.6277461855433060659738094759197823268$

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 264 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not 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$(j)_{-}$
$2$ $11$ $I_{11}$ Split multiplicative -1 1 11 11
$5$ $3$ $IV$ Additive -1 2 4 0
$7$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

## 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
$2$ 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]

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 add nonsplit ordinary ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary 4 1,1 - 1 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

An entry - indicates that the invariants are not computed because the reduction is additive.

## Isogenies

This curve has no rational isogenies. Its isogeny class 350.d 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 $K$ $3$ 3.1.200.1 $$\Z/2\Z$$ Not in database $6$ 6.0.320000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ 8.2.52509870000.1 $$\Z/3\Z$$ 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.