Properties

Label 98826.p2
Conductor 98826
Discriminant -2266747861853082876455402377344
j-invariant \( -\frac{4095503324447959733993040498942529}{2266747861853082876455402377344} \)
CM no
Rank 1
Torsion Structure \(\Z/{7}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -3333198596, 103601868154512]) # or
 
sage: E = EllipticCurve("98826t1")
 
gp: E = ellinit([1, 0, 0, -3333198596, 103601868154512]) \\ or
 
gp: E = ellinit("98826t1")
 
magma: E := EllipticCurve([1, 0, 0, -3333198596, 103601868154512]); // or
 
magma: E := EllipticCurve("98826t1");
 

\( y^2 + x y = x^{3} - 3333198596 x + 103601868154512 \)

Mordell-Weil group structure

\(\Z\times \Z/{7}\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(18382, 6958042\right) \)
\(\hat{h}(P)\) ≈  1.9838124662775718

Torsion generators

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

\( \left(26572, 5799976\right) \)

Integral points

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

\( \left(-42224, 13023556\right) \), \( \left(-42224, -12981332\right) \), \( \left(18382, 6958042\right) \), \( \left(18382, -6976424\right) \), \( \left(26572, 5799976\right) \), \( \left(26572, -5826548\right) \), \( \left(90454, 23239762\right) \), \( \left(90454, -23330216\right) \), \( \left(132412, 44473912\right) \), \( \left(132412, -44606324\right) \), \( \left(2390206, 3693069016\right) \), \( \left(2390206, -3695459222\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 98826 \)  =  \(2 \cdot 3 \cdot 7 \cdot 13 \cdot 181\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-2266747861853082876455402377344 \)  =  \(-1 \cdot 2^{7} \cdot 3^{21} \cdot 7^{7} \cdot 13^{7} \cdot 181^{2} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{4095503324447959733993040498942529}{2266747861853082876455402377344} \)  =  \(-1 \cdot 2^{-7} \cdot 3^{-21} \cdot 7^{-7} \cdot 13^{-7} \cdot 83^{3} \cdot 181^{-2} \cdot 337^{3} \cdot 547^{3} \cdot 10457^{3}\)
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: \(1.98381246628\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.0240899820498\)
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: \( 14406 \)  = \( 7\cdot( 3 \cdot 7 )\cdot7\cdot7\cdot2 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(7\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 98826.2.a.p

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);
 

\( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - q^{10} + 5q^{11} + q^{12} + q^{13} + q^{14} - q^{15} + q^{16} - 3q^{17} + q^{18} - q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 169645056
\( \Gamma_0(N) \)-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);
 

\( L'(E,1) \) ≈ \( 14.0502619706 \)

Local data

This elliptic curve is 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\((j)_{-}\)
\(2\) \(7\) \( I_{7} \) Split multiplicative -1 1 7 7
\(3\) \(21\) \( I_{21} \) Split multiplicative -1 1 21 21
\(7\) \(7\) \( I_{7} \) Split multiplicative -1 1 7 7
\(13\) \(7\) \( I_{7} \) Split multiplicative -1 1 7 7
\(181\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2

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.1.1

$p$-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 47 181
Reduction type split split ordinary split ordinary split ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary nonsplit
$\lambda$-invariant(s) 3 4 1 4 1 2 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 0

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 7.
Its isogeny class 98826.p 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 $E(\Q)_{\rm tors}$ $\cong \Z/{7}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.2184.1 \(\Z/14\Z\) Not in database
6 6.0.10417365504.1 \(\Z/2\Z \times \Z/14\Z\) 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.