Properties

Label 75150.h1
Conductor 75150
Discriminant 1733411074218750000000
j-invariant \( \frac{156406207396688718841}{152178750000000} \)
CM no
Rank 2
Torsion Structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 0, -25255692, 48817755216]); // or
 
magma: E := EllipticCurve("75150i2");
 
sage: E = EllipticCurve([1, -1, 0, -25255692, 48817755216]) # or
 
sage: E = EllipticCurve("75150i2")
 
gp: E = ellinit([1, -1, 0, -25255692, 48817755216]) \\ or
 
gp: E = ellinit("75150i2")
 

\( y^2 + x y = x^{3} - x^{2} - 25255692 x + 48817755216 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(2529, 32148\right) \)\( \left(2829, 723\right) \)
\(\hat{h}(P)\) ≈  3.034312823874.29288073325

Torsion generators

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

\( \left(\frac{11331}{4}, -\frac{11331}{8}\right) \)

Integral points

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

\( \left(2529, 32148\right) \), \( \left(2829, 723\right) \), \( \left(6739, 426318\right) \), \( \left(8685, 691731\right) \), \( \left(319239, 180191943\right) \)

Note: only one of each pair $\pm P$ is listed.

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: \(1733411074218750000000 \)  =  \(2^{7} \cdot 3^{12} \cdot 5^{16} \cdot 167 \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( \frac{156406207396688718841}{152178750000000} \)  =  \(2^{-7} \cdot 3^{-6} \cdot 5^{-10} \cdot 29^{3} \cdot 167^{-1} \cdot 185789^{3}\)
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: \(12.7220018319\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.148374731415\)
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: \( 16 \)  = \( 1\cdot2^{2}\cdot2^{2}\cdot1 \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(2\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 75150.2.a.h

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)
 

\( q - q^{2} + q^{4} - 2q^{7} - q^{8} - 2q^{13} + 2q^{14} + q^{16} - 6q^{17} + 4q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 5806080
\( \Gamma_0(N) \)-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])
 

\( L^{(2)}(E,1)/2! \) ≈ \( 7.5504944195 \)

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\((j)_{-}\)
\(2\) \(1\) \( I_{7} \) Non-split multiplicative 1 1 7 7
\(3\) \(4\) \( I_6^{*} \) Additive -1 2 12 6
\(5\) \(4\) \( I_10^{*} \) Additive 1 2 16 10
\(167\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1

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.

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
\(2\) 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 ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary nonsplit
$\lambda$-invariant(s) 4 - - 2 2,2 2 2 2 2 2,2 2 2 2 2 2 2
$\mu$-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 \(d=\) 2.
Its isogeny class 75150.h consists of 2 curves linked by isogenies of degree 2.

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 \(\Q(\sqrt{334}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
4 4.0.1202400.3 \(\Z/4\Z\) 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.