Properties

Label 100800pd1
Conductor 100800
Discriminant -180592312320000
j-invariant \( \frac{2595575}{1512} \)
CM no
Rank 2
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, 14100, -52400]) # or
 
sage: E = EllipticCurve("100800pd1")
 
gp: E = ellinit([0, 0, 0, 14100, -52400]) \\ or
 
gp: E = ellinit("100800pd1")
 
magma: E := EllipticCurve([0, 0, 0, 14100, -52400]); // or
 
magma: E := EllipticCurve("100800pd1");
 

\( y^2 = x^{3} + 14100 x - 52400 \)

Mordell-Weil group structure

\(\Z^2\)

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(5, 135\right) \)\( \left(26, 576\right) \)
\(\hat{h}(P)\) ≈  1.78816768438210841.1760114055199553

Integral points

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

\((5,\pm 135)\), \((26,\pm 576)\), \((80,\pm 1260)\), \((140,\pm 2160)\), \((144,\pm 2228)\), \((410,\pm 8640)\), \((1050,\pm 34240)\), \((3029,\pm 166833)\), \((1203125,\pm 1319672385)\)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 100800 \)  =  \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-180592312320000 \)  =  \(-1 \cdot 2^{21} \cdot 3^{9} \cdot 5^{4} \cdot 7 \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{2595575}{1512} \)  =  \(2^{-3} \cdot 3^{-3} \cdot 5^{2} \cdot 7^{-1} \cdot 47^{3}\)
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: \(0.663084827052\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.336495494267\)
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: \( 48 \)  = \( 2^{2}\cdot2^{2}\cdot3\cdot1 \)
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: \(1\) (rounded)

Modular invariants

Modular form 100800.2.a.d

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);
 

\( q - q^{7} - 6q^{11} + q^{13} - 3q^{17} - 4q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 331776
\( \Gamma_0(N) \)-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);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 10.7100027177 \)

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\((j)_{-}\)
\(2\) \(4\) \( I_11^{*} \) Additive -1 6 21 3
\(3\) \(4\) \( I_3^{*} \) Additive -1 2 9 3
\(5\) \(3\) \( IV \) Additive -1 2 4 0
\(7\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1

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
\(3\) 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 add add nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss
$\lambda$-invariant(s) - - - 4 2 2 2 2 2 2 4 2 2 2 2,2
$\mu$-invariant(s) - - - 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 100800pd 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 \(\Q(\sqrt{6}) \) \(\Z/3\Z\) Not in database
3 3.1.4200.1 \(\Z/2\Z\) Not in database
6 6.2.423360000.1 \(\Z/6\Z\) Not in database
6.0.2963520000.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
6.0.6914880000.2 \(\Z/3\Z\) 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.