Properties

Label 235200.x2
Conductor 235200
Discriminant -141675749376000000
j-invariant \( -\frac{2401}{6} \)
CM no
Rank 1
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -80033, 20123937]) # or
 
sage: E = EllipticCurve("235200x1")
 
gp: E = ellinit([0, -1, 0, -80033, 20123937]) \\ or
 
gp: E = ellinit("235200x1")
 
magma: E := EllipticCurve([0, -1, 0, -80033, 20123937]); // or
 
magma: E := EllipticCurve("235200x1");
 

\( y^2 = x^{3} - x^{2} - 80033 x + 20123937 \)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(-359, 1568\right) \)
\(\hat{h}(P)\) ≈  1.4918631385182504

Integral points

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

\((-359,\pm 1568)\), \((10323,\pm 1048404)\)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 235200 \)  =  \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-141675749376000000 \)  =  \(-1 \cdot 2^{19} \cdot 3 \cdot 5^{6} \cdot 7^{8} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{2401}{6} \)  =  \(-1 \cdot 2^{-1} \cdot 3^{-1} \cdot 7^{4}\)
Endomorphism Ring: \(\Z\)
Geometric Endomorphism Ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Rank: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(1.49186313852\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.289032245815\)
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: \( 12 \)  = \( 2^{2}\cdot1\cdot1\cdot3 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(1\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 235200.2.a.x

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);
 

\( q - q^{3} + q^{9} - 5q^{11} + 4q^{17} - 8q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 2257920
\( \Gamma_0(N) \)-optimal: yes
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);
 

\( L'(E,1) \) ≈ \( 5.17435864049 \)

Local data

This elliptic curve is not 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\((j)_{-}\)
\(2\) \(4\) \( I_9^{*} \) Additive 1 6 19 1
\(3\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1
\(5\) \(1\) \( I_0^{*} \) Additive 1 2 6 0
\(7\) \(3\) \( IV^{*} \) Additive 1 2 8 0

Galois representations

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

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
\(7\) B.6.3

$p$-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 \(d=\) 7.
Its isogeny class 235200.x consists of 2 curves linked by isogenies of degree 7.

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
3 3.1.1176.1 \(\Z/2\Z\) Not in database
6 6.0.33191424.2 \(\Z/2\Z \times \Z/2\Z\) Not in database
6.6.153664000.1 \(\Z/7\Z\) 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.