Properties

Label 650.f2
Conductor $650$
Discriminant $-85196800$
j-invariant \( -\frac{9836106385}{3407872} \)
CM no
Rank $1$
Torsion structure trivial

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -130, -780])
 
gp: E = ellinit([1, 1, 0, -130, -780])
 
magma: E := EllipticCurve([1, 1, 0, -130, -780]);
 

\(y^2+xy=x^3+x^2-130x-780\)  Toggle raw display

Mordell-Weil group structure

$\Z$

Infinite order Mordell-Weil generator and height

sage: E.gens()
 
magma: Generators(E);
 

$P$ =  \(\left(84, 726\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $1.3951136989889534675481147885$

Integral points

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

\( \left(84, 726\right) \), \( \left(84, -810\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 650 \)  =  $2 \cdot 5^{2} \cdot 13$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-85196800 $  =  $-1 \cdot 2^{18} \cdot 5^{2} \cdot 13 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{9836106385}{3407872} \)  =  $-1 \cdot 2^{-18} \cdot 5 \cdot 7^{3} \cdot 13^{-1} \cdot 179^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.23307943064479354152785074509\dots$
Stable Faltings height: $-0.035160221427556520905609143781\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1.3951136989889534675481147885\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.69411006944000237725327876873\dots$
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: $ 2 $  = $ 2\cdot1\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: $1$ (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) $ ≈ $ 1.9367249329638421313827318380971582290 $

Modular invariants

Modular form   650.2.a.f

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 360
$ \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$(j)_{-}$
$2$ $2$ $I_{18}$ Non-split multiplicative 1 1 18 18
$5$ $1$ $II$ Additive 1 2 2 0
$13$ $1$ $I_{1}$ Non-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 3.4.0.1

$p$-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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type nonsplit ordinary add ordinary ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary
$\lambda$-invariant(s) 1 3 - 1 1 1 1 1 1 1 1 3 1,1 3 1
$\mu$-invariant(s) 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 non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 650.f 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{5}) \) \(\Z/3\Z\) Not in database
$3$ 3.1.1300.1 \(\Z/2\Z\) Not in database
$6$ 6.0.87880000.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
$6$ 6.0.2409834375.1 \(\Z/3\Z\) Not in database
$6$ 6.2.8450000.1 \(\Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/4\Z\) Not in database
$12$ Deg 12 \(\Z/3\Z \times \Z/3\Z\) Not in database
$12$ Deg 12 \(\Z/2\Z \times \Z/6\Z\) Not in database
$18$ 18.6.9644494470890581329345703125.1 \(\Z/9\Z\) Not in database
$18$ 18.0.9687422424964888173375000000000000.4 \(\Z/6\Z\) 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.