Properties

Label 48400.x2
Conductor 48400
Discriminant -3990972620800
j-invariant \( \frac{34295}{22} \)
CM no
Rank 1
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 0, 3832, 31348]); // or
 
magma: E := EllipticCurve("48400cq1");
 
sage: E = EllipticCurve([0, 1, 0, 3832, 31348]) # or
 
sage: E = EllipticCurve("48400cq1")
 
gp: E = ellinit([0, 1, 0, 3832, 31348]) \\ or
 
gp: E = ellinit("48400cq1")
 

\( y^2 = x^{3} + x^{2} + 3832 x + 31348 \)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(7, 242\right) \)
\(\hat{h}(P)\) ≈  1.24372732275

Integral points

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

\((7,\pm 242)\)

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 48400 \)  =  \(2^{4} \cdot 5^{2} \cdot 11^{2}\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(-3990972620800 \)  =  \(-1 \cdot 2^{13} \cdot 5^{2} \cdot 11^{7} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( \frac{34295}{22} \)  =  \(2^{-1} \cdot 5 \cdot 11^{-1} \cdot 19^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(1\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(1.24372732275\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.487681472064\)
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: \( 8 \)  = \( 2\cdot1\cdot2^{2} \)
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: \(1\) (exact)

Modular invariants

Modular form 48400.2.a.x

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)
 

\( q - 2q^{3} + 4q^{7} + q^{9} + 5q^{13} - 7q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 103680
\( \Gamma_0(N) \)-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])
 

\( L'(E,1) \) ≈ \( 4.85234217283 \)

Local data

This elliptic curve is not semistable.

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\((j)_{-}\)
\(2\) \(2\) \( I_5^{*} \) Additive -1 4 13 1
\(5\) \(1\) \( II \) Additive 1 2 2 0
\(11\) \(4\) \( I_1^{*} \) Additive -1 2 7 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]
 

Note: \(p\)-adic regulator data only exists for primes \(p\ge5\) of good ordinary reduction.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type add ordinary add ordinary add ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss
$\lambda$-invariant(s) - 3 - 1 - 1 1,1 1 1 3 3 1 1 1 1,1
$\mu$-invariant(s) - 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 48400.x 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 \(\Q(\sqrt{55}) \) \(\Z/3\Z\) Not in database
3 3.1.2200.1 \(\Z/2\Z\) Not in database
6 6.2.4259200000.1 \(\Z/6\Z\) Not in database
6.0.425920000.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
6.0.3478701600000.2 \(\Z/3\Z\) 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.