Properties

Label 130.a3
Conductor $130$
Discriminant $-10562500$
j-invariant \( -\frac{217081801}{10562500} \)
CM no
Rank $1$
Torsion structure \(\Z/{6}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -13, 156])
 
gp: E = ellinit([1, 0, 1, -13, 156])
 
magma: E := EllipticCurve([1, 0, 1, -13, 156]);
 

\(y^2+xy+y=x^3-13x+156\)  Toggle raw display

Mordell-Weil group structure

$\Z\times \Z/{6}\Z$

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(5, 12\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $0.58523207679736111555815040016$

Torsion generators

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

\( \left(10, 27\right) \)  Toggle raw display

Integral points

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

\( \left(-5, 12\right) \), \( \left(-5, -8\right) \), \( \left(-3, 14\right) \), \( \left(-3, -12\right) \), \( \left(0, 12\right) \), \( \left(0, -13\right) \), \( \left(5, 12\right) \), \( \left(5, -18\right) \), \( \left(6, 14\right) \), \( \left(6, -21\right) \), \( \left(10, 27\right) \), \( \left(10, -38\right) \), \( \left(25, 112\right) \), \( \left(25, -138\right) \), \( \left(75, 612\right) \), \( \left(75, -688\right) \), \( \left(140, 1587\right) \), \( \left(140, -1728\right) \), \( \left(1045, 33262\right) \), \( \left(1045, -34308\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 130 \)  =  $2 \cdot 5 \cdot 13$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-10562500 $  =  $-1 \cdot 2^{2} \cdot 5^{6} \cdot 13^{2} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{217081801}{10562500} \)  =  $-1 \cdot 2^{-2} \cdot 5^{-6} \cdot 13^{-2} \cdot 601^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.027716848423581394600829886950\dots$
Stable Faltings height: $0.027716848423581394600829886950\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.58523207679736111555815040016\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $1.8921144996833156027514115762\dots$
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: $ 24 $  = $ 2\cdot( 2 \cdot 3 )\cdot2 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $6$
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) $ ≈ $ 0.73821739879204444066176003316110218404 $

Modular invariants

Modular form   130.2.a.a

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);
 

\( q - q^{2} - 2q^{3} + q^{4} + q^{5} + 2q^{6} - 4q^{7} - q^{8} + q^{9} - q^{10} - 6q^{11} - 2q^{12} + q^{13} + 4q^{14} - 2q^{15} + q^{16} - 6q^{17} - q^{18} + 2q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 48
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is 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$(j)_{-}$
$2$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$5$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$13$ $2$ $I_{2}$ 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
$2$ 2B 4.12.0.12
$3$ 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]
 

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type nonsplit ordinary split ordinary ordinary split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 2 1 2 1 1 2 1 1 1 1 1 1 1 3 1
$\mu$-invariant(s) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 130.a consists of 4 curves linked by isogenies of degrees dividing 6.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-1}) \) \(\Z/2\Z \times \Z/6\Z\) 2.0.4.1-8450.5-a1
$4$ 4.2.16900.1 \(\Z/12\Z\) Not in database
$4$ 4.0.1040.1 \(\Z/2\Z \times \Z/12\Z\) Not in database
$6$ 6.0.12338352.2 \(\Z/3\Z \times \Z/6\Z\) Not in database
$8$ 8.0.4569760000.4 \(\Z/4\Z \times \Z/12\Z\) Not in database
$9$ 9.3.140541540750000.5 \(\Z/18\Z\) Not in database
$12$ Deg 12 \(\Z/6\Z \times \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/24\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/24\Z\) Not in database
$18$ 18.0.323615533897873990656000000000000.3 \(\Z/2\Z \times \Z/18\Z\) 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.