Properties

Label 265200w1
Conductor 265200
Discriminant -139404395520000
j-invariant \( -\frac{99546915625}{54454842} \)
CM no
Rank 2
Torsion Structure \(\mathrm{Trivial}\)

Related objects

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Minimal Weierstrass equation

magma: E := EllipticCurve([0, -1, 0, -13208, 819312]); // or
 
magma: E := EllipticCurve("265200w1");
 
sage: E = EllipticCurve([0, -1, 0, -13208, 819312]) # or
 
sage: E = EllipticCurve("265200w1")
 
gp: E = ellinit([0, -1, 0, -13208, 819312]) \\ or
 
gp: E = ellinit("265200w1")
 

\( y^2 = x^{3} - x^{2} - 13208 x + 819312 \)

Mordell-Weil group structure

\(\Z^2\)

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(-28, 1080\right) \)\( \left(52, 520\right) \)
\(\hat{h}(P)\) ≈  0.7610569214470.429777345042

Integral points

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

\((-78,\pm 1170)\), \((-28,\pm 1080)\), \((13,\pm 806)\), \((26,\pm 702)\), \((52,\pm 520)\), \((82,\pm 530)\), \((196,\pm 2392)\), \((377,\pm 7020)\), \((1092,\pm 35880)\), \((2132,\pm 98280)\), \((13418,\pm 1554174)\)

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 265200 \)  =  \(2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(-139404395520000 \)  =  \(-1 \cdot 2^{13} \cdot 3^{6} \cdot 5^{4} \cdot 13^{3} \cdot 17 \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( -\frac{99546915625}{54454842} \)  =  \(-1 \cdot 2^{-1} \cdot 3^{-6} \cdot 5^{5} \cdot 13^{-3} \cdot 17^{-1} \cdot 317^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(2\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(0.288154851194\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.540668904836\)
magma: TamagawaNumbers(E);
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Tamagawa product: \( 72 \)  = \( 2^{2}\cdot2\cdot3\cdot3\cdot1 \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(1\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 265200.2.a.w

magma: ModularForm(E);
 
sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 

\( q - q^{3} - 2q^{7} + q^{9} - 3q^{11} + q^{13} - q^{17} - 2q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 705024
\( \Gamma_0(N) \)-optimal: yes
Manin constant: 1

Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 

\( L^{(2)}(E,1)/2! \) ≈ \( 11.2173384829 \)

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(4\) \( I_5^{*} \) Additive -1 4 13 1
\(3\) \(2\) \( I_{6} \) Non-split multiplicative 1 1 6 6
\(5\) \(3\) \( IV \) Additive -1 2 4 0
\(13\) \(3\) \( I_{3} \) Split multiplicative -1 1 3 3
\(17\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 
sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 

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

prime Image of Galois representation
\(3\) B

$p$-adic data

$p$-adic regulators

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

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

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 265200w consists of 2 curves linked by isogenies of degree 3.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 \(\Q(\sqrt{-1}) \) \(\Z/3\Z\) Not in database
3 3.1.44200.1 \(\Z/2\Z\) Not in database
6 6.2.360810720000.7 \(\Z/3\Z\) Not in database
6.0.31258240000.1 \(\Z/6\Z\) Not in database
6.0.3454035520000.1 \(\Z/2\Z \times \Z/2\Z\) 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.