Label 236992.d1
Conductor $236992$
Discriminant $-1.257\times 10^{17}$
j-invariant \( -\frac{621000}{49} \)
CM no
Rank $1$
Torsion structure trivial

Related objects


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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -243340, -49252016])
gp: E = ellinit([0, 0, 0, -243340, -49252016])
magma: E := EllipticCurve([0, 0, 0, -243340, -49252016]);

\(y^2=x^3-243340x-49252016\)  Toggle raw display

Mordell-Weil group structure


Infinite order Mordell-Weil generator and height

sage: E.gens()
magma: Generators(E);

$P$ =  \(\left(1058, 29624\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $1.4650688422639082853983351794$

Integral points

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

\((1058,\pm 29624)\)  Toggle raw display


sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
Conductor: \( 236992 \)  =  $2^{6} \cdot 7 \cdot 23^{2}$
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
Discriminant: $-125738623918702592 $  =  $-1 \cdot 2^{15} \cdot 7^{2} \cdot 23^{8} $
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
j-invariant: \( -\frac{621000}{49} \)  =  $-1 \cdot 2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 7^{-2} \cdot 23$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.0281139226991114754705829528\dots$
Stable Faltings height: $-0.92864953028691995517212575357\dots$

BSD invariants

sage: E.rank()
magma: Rank(E);
Analytic rank: $1$
sage: E.regulator()
magma: Regulator(E);
Regulator: $1.4650688422639082853983351794\dots$
sage: E.period_lattice().omega()
magma: RealPeriod(E);
Real period: $0.10699308576496095318302841600\dots$
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: $ 24 $  = $ 2^{2}\cdot2\cdot3 $
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: $1$ (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) $ ≈ $ 3.7620536710059453520508045696660922046 $

Modular invariants

Modular form 236992.2.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);

\( q - 3q^{3} + q^{7} + 6q^{9} - 6q^{13} + 7q^{17} + 5q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
magma: ModularDegree(E);
Modular degree: 4733952
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 3 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$(j)_{-}$
$2$ $4$ $I_5^{*}$ Additive 1 6 15 0
$7$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$23$ $3$ $IV^{*}$ Additive -1 2 8 0

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
$2$ 2G

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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.


This curve has no rational isogenies. Its isogeny class 236992.d consists of this curve only.

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
$3$ 3.1.4232.1 \(\Z/2\Z\) Not in database
$6$ 6.0.143278592.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
$8$ Deg 8 \(\Z/3\Z\) Not in database
$12$ Deg 12 \(\Z/4\Z\) 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.