Properties

Label 510.g1
Conductor $510$
Discriminant $2210850$
j-invariant \( \frac{15916310615119911121}{2210850} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -52405, -4621873]) # or
 
sage: E = EllipticCurve("510g4")
 
gp: E = ellinit([1, 0, 0, -52405, -4621873]) \\ or
 
gp: E = ellinit("510g4")
 
magma: E := EllipticCurve([1, 0, 0, -52405, -4621873]); // or
 
magma: E := EllipticCurve("510g4");
 

\( y^2 + x y = x^{3} - 52405 x - 4621873 \)

Mordell-Weil group structure

\(\Z/{2}\Z\)

Torsion generators

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

\( \left(-\frac{529}{4}, \frac{529}{8}\right) \)

Integral points

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

None

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 510 \)  =  \(2 \cdot 3 \cdot 5 \cdot 17\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(2210850 \)  =  \(2 \cdot 3^{2} \cdot 5^{2} \cdot 17^{3} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{15916310615119911121}{2210850} \)  =  \(2^{-1} \cdot 3^{-2} \cdot 5^{-2} \cdot 17^{-3} \cdot 23^{3} \cdot 109367^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Rank: \(0\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(1\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.315541386996\)
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: \( 4 \)  = \( 1\cdot2\cdot2\cdot1 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(2\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(9\) (exact)

Modular invariants

Modular form   510.2.a.g

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 864
\( \Gamma_0(N) \)-optimal: no
Manin constant: 1

Special L-value

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);
 

\( L(E,1) \) ≈ \( 2.83987248296 \)

Local data

This elliptic curve is semistable.

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\((j)_{-}\)
\(2\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1
\(3\) \(2\) \( I_{2} \) Split multiplicative -1 1 2 2
\(5\) \(2\) \( I_{2} \) Split multiplicative -1 1 2 2
\(17\) \(1\) \( I_{3} \) Non-split multiplicative 1 1 3 3

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X6.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right)$ and has index 3.

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 mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime Image of Galois representation
\(2\) B
\(3\) B.1.2

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]
 

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$ 2 3 5 17
Reduction type split split split nonsplit
$\lambda$-invariant(s) 4 3 1 0
$\mu$-invariant(s) 0 1 0 0

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

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 510.g consists of 4 curves linked by isogenies of degrees dividing 6.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-3}) \) \(\Z/6\Z\) 2.0.3.1-86700.1-g4
$2$ \(\Q(\sqrt{34}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$3$ 3.1.24300.1 \(\Z/6\Z\) Not in database
$4$ 4.0.122400.2 \(\Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-3}, \sqrt{34})\) \(\Z/2\Z \times \Z/6\Z\) Not in database
$6$ 6.0.1771470000.3 \(\Z/3\Z \times \Z/6\Z\) Not in database
$6$ \( x^{6} - 102 x^{4} - 120 x^{3} + 3468 x^{2} - 12240 x - 35704 \) \(\Z/2\Z \times \Z/6\Z\) Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.