Properties

Label 235200.ux7
Conductor 235200
Discriminant 2914472558592000000000
j-invariant \( \frac{13619385906841}{6048000} \)
CM no
Rank 2
Torsion Structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -39005633, 93715660863]) # or
 
sage: E = EllipticCurve("235200ux1")
 
gp: E = ellinit([0, 1, 0, -39005633, 93715660863]) \\ or
 
gp: E = ellinit("235200ux1")
 
magma: E := EllipticCurve([0, 1, 0, -39005633, 93715660863]); // or
 
magma: E := EllipticCurve("235200ux1");
 

\( y^2 = x^{3} + x^{2} - 39005633 x + 93715660863 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(-1457, 384000\right) \)\( \left(79, 301056\right) \)
\(\hat{h}(P)\) ≈  1.75009047306830381.9052699958466284

Torsion generators

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

\( \left(3663, 0\right) \)

Integral points

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

\((-5582,\pm 370875)\), \((-3743,\pm 432768)\), \((-1457,\pm 384000)\), \((79,\pm 301056)\), \((3313,\pm 29400)\), \((3418,\pm 18375)\), \( \left(3663, 0\right) \), \((4293,\pm 73500)\), \((40527,\pm 8067072)\), \((200293,\pm 89596500)\)

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: \(2914472558592000000000 \)  =  \(2^{26} \cdot 3^{3} \cdot 5^{9} \cdot 7^{7} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{13619385906841}{6048000} \)  =  \(2^{-8} \cdot 3^{-3} \cdot 5^{-3} \cdot 7^{-1} \cdot 11^{3} \cdot 13^{3} \cdot 167^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Rank: \(2\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(2.45612193147\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.140638137036\)
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: \( 192 \)  = \( 2^{2}\cdot3\cdot2^{2}\cdot2^{2} \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(2\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 235200.2.a.ux

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);
 

\( q + q^{3} + q^{9} - 2q^{13} - 6q^{17} - 8q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 21233664
\( \Gamma_0(N) \)-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);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 16.5803718132 \)

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\((j)_{-}\)
\(2\) \(4\) \( I_16^{*} \) Additive -1 6 26 8
\(3\) \(3\) \( I_{3} \) Split multiplicative -1 1 3 3
\(5\) \(4\) \( I_3^{*} \) Additive 1 2 9 3
\(7\) \(4\) \( I_1^{*} \) Additive -1 2 7 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 X13.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$ and has index 6.

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
\(2\) B
\(3\) 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 \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 235200.ux consists of 8 curves linked by isogenies of degrees dividing 12.

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 \(\Q(\sqrt{14}) \) \(\Z/4\Z\) Not in database
\(\Q(\sqrt{105}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
\(\Q(\sqrt{30}) \) \(\Z/4\Z\) Not in database
\(\Q(\sqrt{70}) \) \(\Z/6\Z\) Not in database
4 \(\Q(\sqrt{5}, \sqrt{14})\) \(\Z/12\Z\) Not in database
\(\Q(\sqrt{14}, \sqrt{30})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
\(\Q(\sqrt{6}, \sqrt{70})\) \(\Z/2\Z \times \Z/6\Z\) Not in database
\(\Q(\sqrt{21}, \sqrt{30})\) \(\Z/12\Z\) Not in database
6 6.0.116169984000.9 \(\Z/6\Z\) 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.