Properties

Label 177450.hb7
Conductor 177450
Discriminant 878644392096744000000000
j-invariant \( \frac{1882742462388824401}{11650189824000} \)
CM no
Rank 1
Torsion Structure \(\Z/{4}\Z\)

Related objects

Downloads

Learn more about

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, -108688213, 433752569531]) # or
 
sage: E = EllipticCurve("177450dr1")
 
gp: E = ellinit([1, 1, 1, -108688213, 433752569531]) \\ or
 
gp: E = ellinit("177450dr1")
 
magma: E := EllipticCurve([1, 1, 1, -108688213, 433752569531]); // or
 
magma: E := EllipticCurve("177450dr1");
 

\( y^2 + x y + y = x^{3} + x^{2} - 108688213 x + 433752569531 \)

Mordell-Weil group structure

\(\Z\times \Z/{4}\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(6515, 43992\right) \)
\(\hat{h}(P)\) ≈  1.1841609076708908

Torsion generators

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

\( \left(10015, 586492\right) \)

Integral points

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

\( \left(6375, -3188\right) \), \( \left(6515, 43992\right) \), \( \left(6515, -50508\right) \), \( \left(7051, 130660\right) \), \( \left(7051, -137712\right) \), \( \left(7649, 219762\right) \), \( \left(7649, -227412\right) \), \( \left(10015, 586492\right) \), \( \left(10015, -596508\right) \), \( \left(16775, 1816812\right) \), \( \left(16775, -1833588\right) \), \( \left(25975, 3877612\right) \), \( \left(25975, -3903588\right) \), \( \left(101015, 31890492\right) \), \( \left(101015, -31991508\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 177450 \)  =  \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(878644392096744000000000 \)  =  \(2^{12} \cdot 3^{6} \cdot 5^{9} \cdot 7^{4} \cdot 13^{7} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{1882742462388824401}{11650189824000} \)  =  \(2^{-12} \cdot 3^{-6} \cdot 5^{-3} \cdot 7^{-4} \cdot 13^{-1} \cdot 23^{3} \cdot 37^{3} \cdot 1451^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Rank: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(1.18416090767\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.0892469517521\)
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: \( 1536 \)  = \( ( 2^{2} \cdot 3 )\cdot2\cdot2^{2}\cdot2^{2}\cdot2^{2} \)
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: \(1\) (exact)

Modular invariants

Modular form 177450.2.a.hb

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);
 

\( q + q^{2} - q^{3} + q^{4} - q^{6} + q^{7} + q^{8} + q^{9} - q^{12} + q^{14} + q^{16} - 6q^{17} + q^{18} + 4q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 37158912
\( \Gamma_0(N) \)-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);
 

\( L'(E,1) \) ≈ \( 10.1455441338 \)

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\((j)_{-}\)
\(2\) \(12\) \( I_{12} \) Split multiplicative -1 1 12 12
\(3\) \(2\) \( I_{6} \) Non-split multiplicative 1 1 6 6
\(5\) \(4\) \( I_3^{*} \) Additive 1 2 9 3
\(7\) \(4\) \( I_{4} \) Split multiplicative -1 1 4 4
\(13\) \(4\) \( I_1^{*} \) Additive 1 2 7 1

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 X13h.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 3 \end{array}\right)$ and has index 12.

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
\(2\) B
\(3\) B

$p$-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, 3, 4, 6 and 12.
Its isogeny class 177450.hb consists of 8 curves linked by isogenies of degrees dividing 12.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 \(\Q(\sqrt{65}) \) \(\Z/2\Z \times \Z/12\Z\) Not in database
4 4.4.861224000.1 \(\Z/8\Z\) Not in database
6 6.0.3008726413875.1 \(\Z/12\Z\) 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.