Properties

 Label 78400.jd1 Conductor $78400$ Discriminant $4.390\times 10^{14}$ j-invariant $$\frac{2185454}{625}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -24033, 1027937])

gp: E = ellinit([0, -1, 0, -24033, 1027937])

magma: E := EllipticCurve([0, -1, 0, -24033, 1027937]);

Simplified equation

 $$y^2=x^3-x^2-24033x+1027937$$ y^2=x^3-x^2-24033x+1027937 (homogenize, simplify) $$y^2z=x^3-x^2z-24033xz^2+1027937z^3$$ y^2z=x^3-x^2z-24033xz^2+1027937z^3 (dehomogenize, simplify) $$y^2=x^3-1946700x+743526000$$ y^2=x^3-1946700x+743526000 (homogenize, minimize)

Mordell-Weil group structure

$$\Z/{2}\Z$$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(47, 0\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(47, 0\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$78400$$ = $2^{6} \cdot 5^{2} \cdot 7^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $439040000000000$ = $2^{17} \cdot 5^{10} \cdot 7^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{2185454}{625}$$ = $2 \cdot 5^{-4} \cdot 103^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.5161877934083862211857214672\dots$ Stable Faltings height: $-0.75696720586574814739874189067\dots$

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: $0.49211305033473037736187526961\dots$ 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: $16$  = $2\cdot2^{2}\cdot2$ 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: $L(E,1)$ ≈ $1.9684522013389215094475010784$

Modular invariants

Modular form 78400.2.a.jd

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);

$$q + 2 q^{3} + q^{9} - 4 q^{11} + 2 q^{13} - 2 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 294912 $\Gamma_0(N)$-optimal: no 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$(j)_{-}$
$2$ $2$ $I_{7}^{*}$ Additive 1 6 17 0
$5$ $4$ $I_{4}^{*}$ Additive 1 2 10 4
$7$ $2$ $III$ Additive -1 2 3 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
$2$ 2B 8.6.0.3

$p$-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

$p$ Reduction type $\lambda$-invariant(s) 2 5 7 add add add - - - - - -

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ 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.
Its isogeny class 78400.jd consists of 2 curves linked by isogenies of degree 2.

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:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{14})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $4$ 4.0.68600.1 $$\Z/4\Z$$ Not in database $8$ 8.4.1233639178240000.54 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.301181440000.1 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ 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.