Properties

Label 75150.t1
Conductor 75150
Discriminant -51360328125000
j-invariant \( -\frac{62394574179743883}{167000} \)
CM no
Rank 1
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 0, -5577567, -5068691659]); // or
 
magma: E := EllipticCurve("75150b2");
 
sage: E = EllipticCurve([1, -1, 0, -5577567, -5068691659]) # or
 
sage: E = EllipticCurve("75150b2")
 
gp: E = ellinit([1, -1, 0, -5577567, -5068691659]) \\ or
 
gp: E = ellinit("75150b2")
 

\( y^2 + x y = x^{3} - x^{2} - 5577567 x - 5068691659 \)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(\frac{13311029}{361}, \frac{48335207787}{6859}\right) \)
\(\hat{h}(P)\) ≈  13.6238924705

Integral points

magma: IntegralPoints(E);
 
sage: E.integral_points()
 
None

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 75150 \)  =  \(2 \cdot 3^{2} \cdot 5^{2} \cdot 167\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(-51360328125000 \)  =  \(-1 \cdot 2^{3} \cdot 3^{9} \cdot 5^{9} \cdot 167 \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( -\frac{62394574179743883}{167000} \)  =  \(-1 \cdot 2^{-3} \cdot 3^{3} \cdot 5^{-3} \cdot 7^{3} \cdot 11^{3} \cdot 17^{3} \cdot 101^{3} \cdot 167^{-1}\)
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: \(13.6238924705\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.0491199997946\)
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: \( 8 \)  = \( 1\cdot2\cdot2^{2}\cdot1 \)
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: \(1\) (exact)

Modular invariants

Modular form 75150.2.a.t

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)
 

\( q - q^{2} + q^{4} + q^{7} - q^{8} + 4q^{13} - q^{14} + q^{16} + 5q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 1244160
\( \Gamma_0(N) \)-optimal: no
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])
 

\( L'(E,1) \) ≈ \( 5.35364476284 \)

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\((j)_{-}\)
\(2\) \(1\) \( I_{3} \) Non-split multiplicative 1 1 3 3
\(3\) \(2\) \( III^{*} \) Additive 1 2 9 0
\(5\) \(4\) \( I_3^{*} \) Additive 1 2 9 3
\(167\) \(1\) \( I_{1} \) 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.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 167
Reduction type nonsplit add add ordinary ss ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary nonsplit
$\lambda$-invariant(s) 4 - - 1 1,1 1 1,1 1 1 1 1 1 1 1 1 1
$\mu$-invariant(s) 0 - - 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 75150.t 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 \(\Q(\sqrt{-15}) \) \(\Z/3\Z\) Not in database
3 3.1.20040.1 \(\Z/2\Z\) Not in database
6 \( x^{6} - 3 x^{5} - 329 x^{3} + 501 x^{2} - 1506 x + 28556 \) \(\Z/3\Z\) Not in database
6.0.8048096064000.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
6.0.6024024000.1 \(\Z/6\Z\) 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.