Properties

Label 4025.g1
Conductor $4025$
Discriminant $3081640625$
j-invariant \( \frac{476196576129}{197225} \)
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, -4067, -98784])
 
gp: E = ellinit([1, -1, 0, -4067, -98784])
 
magma: E := EllipticCurve([1, -1, 0, -4067, -98784]);
 

Minimal equation

Minimal equation

Simplified equation

\(y^2+xy=x^3-x^2-4067x-98784\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3-x^2z-4067xz^2-98784z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-65075x-6387250\) Copy content Toggle raw display (homogenize, minimize)

Mordell-Weil group structure

\(\Z/{2}\Z\)

Torsion generators

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

\( \left(-36, 18\right) \) Copy content Toggle raw display

Integral points

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

\( \left(-36, 18\right) \) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 4025 \)  =  $5^{2} \cdot 7 \cdot 23$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $3081640625 $  =  $5^{8} \cdot 7^{3} \cdot 23 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{476196576129}{197225} \)  =  $3^{3} \cdot 5^{-2} \cdot 7^{-3} \cdot 19^{3} \cdot 23^{-1} \cdot 137^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.78270253811718301409485586192\dots$
Stable Faltings height: $-0.022016418099867173205523804693\dots$

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: $0.59784252184138565563619952987\dots$
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: $ 12 $  = $ 2^{2}\cdot3\cdot1 $
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) $ ≈ $ 1.7935275655241569669085985896 $

Modular invariants

Modular form   4025.2.a.g

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);
 

\( q + q^{2} - q^{4} + q^{7} - 3 q^{8} - 3 q^{9} + 2 q^{11} - 4 q^{13} + q^{14} - q^{16} + 6 q^{17} - 3 q^{18} - 8 q^{19} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 3456
$ \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$(j)_{-}$
$5$ $4$ $I_{2}^{*}$ Additive 1 2 8 2
$7$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$23$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 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 2.3.0.1

$p$-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 5 7 23
Reduction type ord ss add split nonsplit
$\lambda$-invariant(s) 4 8,2 - 1 0
$\mu$-invariant(s) 0 0,0 - 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.
Its isogeny class 4025.g 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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{161}) \) \(\Z/2\Z \oplus \Z/2\Z\) Not in database
$4$ 4.0.64400.3 \(\Z/4\Z\) Not in database
$8$ 8.0.107503718560000.4 \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ 8.2.9562691671875.4 \(\Z/6\Z\) 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.