Properties

Label 97020l1
Conductor $97020$
Discriminant $-9.375\times 10^{15}$
j-invariant \( -\frac{2239956387422208}{27680640625} \)
CM no
Rank $2$
Torsion structure trivial

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Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -190008, 32217668])
 
gp: E = ellinit([0, 0, 0, -190008, 32217668])
 
magma: E := EllipticCurve([0, 0, 0, -190008, 32217668]);
 

\(y^2=x^3-190008x+32217668\)

Mordell-Weil group structure

\(\Z^2\)

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(344, 2750\right) \)\( \left(\frac{1}{4}, \frac{45375}{8}\right) \)
\(\hat{h}(P)\) ≈  $0.30435705854077376337644747805$$1.2205350831333608140046491130$

Integral points

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

\((-272,\pm 7986)\), \((212,\pm 1210)\), \((256,\pm 594)\), \((344,\pm 2750)\), \((421,\pm 5181)\), \((548,\pm 9626)\), \((844,\pm 21750)\), \((1213,\pm 39831)\), \((9844,\pm 975750)\)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 97020 \)  =  \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-9375100812000000 \)  =  \(-1 \cdot 2^{8} \cdot 3^{3} \cdot 5^{6} \cdot 7^{2} \cdot 11^{6} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{2239956387422208}{27680640625} \)  =  \(-1 \cdot 2^{13} \cdot 3^{6} \cdot 5^{-6} \cdot 7 \cdot 11^{-6} \cdot 13^{3} \cdot 29^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: \(2\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(0.36192198387266150066213379539\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.41132345811452227447329091208\)
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: \( 72 \)  = \( 3\cdot2\cdot2\cdot1\cdot( 2 \cdot 3 ) \)
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 97020.2.a.t

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);
 

\( q - q^{5} + q^{11} - 5q^{13} - 5q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 642816
\( \Gamma_0(N) \)-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);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 10.718424142140347200137677019478694452 \)

Local data

This elliptic curve is not semistable. There are 5 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\((j)_{-}\)
\(2\) \(3\) \(IV^{*}\) Additive -1 2 8 0
\(3\) \(2\) \(III\) Additive 1 2 3 0
\(5\) \(2\) \(I_{6}\) Non-split multiplicative 1 1 6 6
\(7\) \(1\) \(II\) Additive -1 2 2 0
\(11\) \(6\) \(I_{6}\) Split multiplicative -1 1 6 6

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(5,20) if E.conductor().valuation(p)<2]
 

Note: \(p\)-adic regulator data only exists for primes \(p\ge 5\) 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 nonsplit add split ordinary ss ordinary ordinary ss ordinary ordinary ss ordinary ordinary
$\lambda$-invariant(s) - - 2 - 3 2 2,2 2 2 2,2 2 2 2,2 2 2
$\mu$-invariant(s) - - 0 - 0 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 97020l consists of 2 curves linked by isogenies of degree 3.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-7}) \) \(\Z/3\Z\) Not in database
$3$ 3.1.588.1 \(\Z/2\Z\) Not in database
$6$ 6.0.1037232.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
$6$ 6.2.588110544.1 \(\Z/3\Z\) Not in database
$6$ 6.0.2420208.1 \(\Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/4\Z\) Not in database
$12$ Deg 12 \(\Z/3\Z \times \Z/3\Z\) Not in database
$12$ 12.0.52716660869376.1 \(\Z/2\Z \times \Z/6\Z\) Not in database
$18$ 18.0.3067753524267383870305529050285577632047000000000000.2 \(\Z/9\Z\) Not in database
$18$ 18.2.203412153331596396123869184.1 \(\Z/6\Z\) 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.