Properties

 Label 1950m1 Conductor $1950$ Discriminant $-799621875000$ j-invariant $$\frac{7604375}{2047032}$$ CM no Rank $0$ Torsion structure $$\Z/{3}\Z$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, 299, -42952])

gp: E = ellinit([1, 0, 1, 299, -42952])

magma: E := EllipticCurve([1, 0, 1, 299, -42952]);

$$y^2+xy+y=x^3+299x-42952$$

Mordell-Weil group structure

$\Z/{3}\Z$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(52, 311\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(52, 311\right)$$, $$\left(52, -364\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$1950$$ = $2 \cdot 3 \cdot 5^{2} \cdot 13$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-799621875000$ = $-1 \cdot 2^{3} \cdot 3^{9} \cdot 5^{8} \cdot 13$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{7604375}{2047032}$$ = $2^{-3} \cdot 3^{-9} \cdot 5^{4} \cdot 13^{-1} \cdot 23^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.96312336436114588580055054085\dots$ Stable Faltings height: $-0.10983524392825436393328901463\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.42055104756763679612280035122\dots$ 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: $27$  = $1\cdot3^{2}\cdot3\cdot1$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $3$ 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: $L(E,1)$ ≈ $1.2616531427029103883684010536520701950$

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);

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

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 3240 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$3$ $9$ $I_{9}$ Split multiplicative -1 1 9 9
$5$ $3$ $IV^{*}$ Additive -1 2 8 0
$13$ $1$ $I_{1}$ 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
$3$ 3B.1.1 3.8.0.1

$p$-adic regulators

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

All $p$-adic regulators are identically $1$ since the rank is $0$.

Iwasawa invariants

$p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 13 nonsplit split add split 4 1 - 1 0 0 - 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

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

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 1950m consists of 2 curves linked by isogenies of degree 3.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.7800.1 $$\Z/6\Z$$ Not in database $6$ 6.0.18982080000.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database $6$ 6.0.481966875.1 $$\Z/3\Z \times \Z/3\Z$$ Not in database $9$ 9.3.8783846296875.1 $$\Z/9\Z$$ Not in database $12$ Deg 12 $$\Z/12\Z$$ Not in database $18$ 18.0.4959960281582022744768000000000000.1 $$\Z/3\Z \times \Z/6\Z$$ 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.