Properties

Label 98.a2
Conductor $98$
Discriminant $-215886856192$
j-invariant \( -\frac{548347731625}{1835008} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -8355, 291341])
 
gp: E = ellinit([1, 1, 0, -8355, 291341])
 
magma: E := EllipticCurve([1, 1, 0, -8355, 291341]);
 

\(y^2+xy=x^3+x^2-8355x+291341\) Copy content Toggle raw display

Mordell-Weil group structure

$\Z/{2}\Z$

Torsion generators

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

\( \left(-106, 53\right) \) Copy content Toggle raw display

Integral points

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

\( \left(-106, 53\right) \) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 98 \)  =  $2 \cdot 7^{2}$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-215886856192 $  =  $-1 \cdot 2^{18} \cdot 7^{7} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{548347731625}{1835008} \)  =  $-1 \cdot 2^{-18} \cdot 5^{3} \cdot 7^{-1} \cdot 1637^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.0394824646707684554248426658\dots$
Stable Faltings height: $0.066527390143111802872166294078\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $0$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $1.0019771957698767743050371648\dots$
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: $ 4 $  = $ 2\cdot2 $
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: $1$ (exact)
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);
 
Special value: $ L(E,1) $ ≈ $ 1.0019771957698767743050371648 $

Modular invariants

Modular form   98.2.a.a

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);
 

\( q - q^{2} + 2 q^{3} + q^{4} - 2 q^{6} - q^{8} + q^{9} + 2 q^{12} + 4 q^{13} + q^{16} - 6 q^{17} - q^{18} - 2 q^{19} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 144
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 2 primes of bad reduction:

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$(j)_{-}$
$2$ $2$ $I_{18}$ Non-split multiplicative 1 1 18 18
$7$ $2$ $I_{1}^{*}$ Additive -1 2 7 1

Galois representations

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 $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 8.6.0.1
$3$ 3B 9.12.0.1

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
 

All $p$-adic regulators are identically $1$ since the rank is $0$.

Iwasawa invariants

$p$ 2 3 7
Reduction type nonsplit ord add
$\lambda$-invariant(s) 2 0 -
$\mu$-invariant(s) 0 0 -

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

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, 3, 6, 9 and 18.
Its isogeny class 98.a consists of 6 curves linked by isogenies of degrees dividing 18.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 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
$2$ \(\Q(\sqrt{-7}) \) \(\Z/2\Z \times \Z/2\Z\) 2.0.7.1-28.2-a1
$2$ \(\Q(\sqrt{21}) \) \(\Z/6\Z\) 2.2.21.1-28.1-a1
$4$ 4.2.448.1 \(\Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-3}, \sqrt{-7})\) \(\Z/2\Z \times \Z/6\Z\) Not in database
$6$ 6.0.12252303.1 \(\Z/2\Z \times \Z/6\Z\) Not in database
$6$ 6.6.330812181.2 \(\Z/18\Z\) Not in database
$8$ 8.0.120472576.1 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.0.9834496.2 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.4.796594176.2 \(\Z/12\Z\) Not in database
$12$ 12.0.1351070359234281.1 \(\Z/6\Z \times \Z/6\Z\) Not in database
$12$ 12.0.2233402022407689.1 \(\Z/2\Z \times \Z/18\Z\) Not in database
$12$ 12.0.109436699097976761.1 \(\Z/2\Z \times \Z/18\Z\) Not in database
$16$ Deg 16 \(\Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/12\Z\) Not in database
$16$ 16.0.634562281237118976.5 \(\Z/2\Z \times \Z/12\Z\) Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.