Properties

Label 277200.fl3
Conductor $277200$
Discriminant $7.531\times 10^{18}$
j-invariant \( \frac{2177286259681}{161417025} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -972075, 344452250]) # or
 
sage: E = EllipticCurve("277200fl2")
 
gp: E = ellinit([0, 0, 0, -972075, 344452250]) \\ or
 
gp: E = ellinit("277200fl2")
 
magma: E := EllipticCurve([0, 0, 0, -972075, 344452250]); // or
 
magma: E := EllipticCurve("277200fl2");
 

\( y^2 = x^{3} - 972075 x + 344452250 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(2335, -103950\right) \)\( \left(-41, 19602\right) \)
\(\hat{h}(P)\) ≈  $1.0740602801785843$$2.174965581963359$

Torsion generators

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

\( \left(685, 0\right) \), \( \left(445, 0\right) \)

Integral points

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

\( \left(-1130, 0\right) \), \((-899,\pm 22176)\), \((-305,\pm 24750)\), \((-41,\pm 19602)\), \((85,\pm 16200)\), \((245,\pm 11000)\), \((391,\pm 4914)\), \((410,\pm 3850)\), \( \left(445, 0\right) \), \( \left(685, 0\right) \), \((806,\pm 9196)\), \((949,\pm 16632)\), \((1495,\pm 47250)\), \((2335,\pm 103950)\), \((4285,\pm 273600)\), \((7445,\pm 637000)\), \((11245,\pm 1188000)\), \((28570,\pm 4826250)\)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 277200 \)  =  \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(7531072718400000000 \)  =  \(2^{12} \cdot 3^{8} \cdot 5^{8} \cdot 7^{2} \cdot 11^{4} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{2177286259681}{161417025} \)  =  \(3^{-2} \cdot 5^{-2} \cdot 7^{-2} \cdot 11^{-4} \cdot 13^{3} \cdot 997^{3}\)
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);
 
Rank: \(2\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(2.00210245123\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.229790053241\)
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: \( 512 \)  = \( 2^{2}\cdot2^{2}\cdot2^{2}\cdot2\cdot2^{2} \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(4\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 277200.2.a.fl

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);
 

\( q - q^{7} + q^{11} + 2q^{13} - 6q^{17} + 4q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 6291456
\( \Gamma_0(N) \)-optimal: no
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);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 14.7220233236 \)

Local data

This elliptic curve is not semistable.

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\((j)_{-}\)
\(2\) \(4\) \( I_4^{*} \) Additive -1 4 12 0
\(3\) \(4\) \( I_2^{*} \) Additive -1 2 8 2
\(5\) \(4\) \( I_2^{*} \) Additive 1 2 8 2
\(7\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2
\(11\) \(4\) \( I_{4} \) Split multiplicative -1 1 4 4

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X25k.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$ and has index 24.

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
\(2\) Cs

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) 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.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 277200.fl consists of 6 curves linked by isogenies of degrees dividing 8.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{15}) \) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-7}, \sqrt{-15})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{7}, \sqrt{-15})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{11}, \sqrt{15})\) \(\Z/2\Z \times \Z/8\Z\) Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.