Properties

Label 1950y1
Conductor $1950$
Discriminant $-5.747\times 10^{17}$
j-invariant \( -\frac{198417696411528597145}{22989483914821632} \)
CM no
Rank $0$
Torsion structure \(\Z/{5}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -355303, -89334583])
 
gp: E = ellinit([1, 0, 0, -355303, -89334583])
 
magma: E := EllipticCurve([1, 0, 0, -355303, -89334583]);
 

\(y^2+xy=x^3-355303x-89334583\)  Toggle raw display

Mordell-Weil group structure

$\Z/{5}\Z$

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(806, 11765\right) \)  Toggle raw display

Integral points

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

\( \left(806, 11765\right) \), \( \left(806, -12571\right) \), \( \left(2678, 133445\right) \), \( \left(2678, -136123\right) \)  Toggle raw display

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: $-574737097870540800 $  =  $-1 \cdot 2^{20} \cdot 3^{10} \cdot 5^{2} \cdot 13^{5} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{198417696411528597145}{22989483914821632} \)  =  $-1 \cdot 2^{-20} \cdot 3^{-10} \cdot 5 \cdot 13^{-5} \cdot 3410909^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.1440101895424695993789303717\dots$
Stable Faltings height: $1.8757705374701195369454704828\dots$

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: $0.097140536637882061932007556089\dots$
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: $ 1000 $  = $ ( 2^{2} \cdot 5 )\cdot( 2 \cdot 5 )\cdot1\cdot5 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $5$
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) $ ≈ $ 3.8856214655152824772803022435774734078 $

Modular invariants

Modular form   1950.2.a.bb

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 33600
$ \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$(j)_{-}$
$2$ $20$ $I_{20}$ Split multiplicative -1 1 20 20
$3$ $10$ $I_{10}$ Split multiplicative -1 1 10 10
$5$ $1$ $II$ Additive 1 2 2 0
$13$ $5$ $I_{5}$ Split multiplicative -1 1 5 5

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
$5$ 5B.1.1 5.24.0.1

$p$-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

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

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ 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 $d=$ 5.
Its isogeny class 1950y consists of 2 curves linked by isogenies of degree 5.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{5}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.1.1300.1 \(\Z/10\Z\) Not in database
$6$ 6.0.87880000.1 \(\Z/2\Z \times \Z/10\Z\) Not in database
$8$ 8.2.1264873866750000.9 \(\Z/15\Z\) Not in database
$12$ Deg 12 \(\Z/20\Z\) Not in database
$20$ 20.0.4656612873077392578125.1 \(\Z/5\Z \times \Z/5\Z\) 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.