Properties

Label 179088.n1
Conductor 179088
Discriminant -503306095088369664
j-invariant \( -\frac{418288977642645996769}{122877464621184} \)
CM no
Rank 0
Torsion Structure \(\mathrm{Trivial}\)

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Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([0, -1, 0, -2492896, -1514524928]); // or
 
magma: E := EllipticCurve("179088bh1");
 
sage: E = EllipticCurve([0, -1, 0, -2492896, -1514524928]) # or
 
sage: E = EllipticCurve("179088bh1")
 
gp: E = ellinit([0, -1, 0, -2492896, -1514524928]) \\ or
 
gp: E = ellinit("179088bh1")
 

\( y^2 = x^{3} - x^{2} - 2492896 x - 1514524928 \)

Mordell-Weil group structure

Trivial

Integral points

magma: IntegralPoints(E);
 
sage: E.integral_points()
 
None

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 179088 \)  =  \(2^{4} \cdot 3 \cdot 7 \cdot 13 \cdot 41\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(-503306095088369664 \)  =  \(-1 \cdot 2^{19} \cdot 3^{7} \cdot 7^{7} \cdot 13 \cdot 41 \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( -\frac{418288977642645996769}{122877464621184} \)  =  \(-1 \cdot 2^{-7} \cdot 3^{-7} \cdot 7^{-7} \cdot 13^{-1} \cdot 41^{-1} \cdot 43^{3} \cdot 173923^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(0\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(1\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.0600739032781\)
magma: TamagawaNumbers(E);
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Tamagawa product: \( 2 \)  = \( 2\cdot1\cdot1\cdot1\cdot1 \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(1\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(9\) (exact)

Modular invariants

Modular form 179088.2.a.n

magma: ModularForm(E);
 
sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 

\( q - q^{3} - q^{5} - q^{7} + q^{9} + 2q^{11} - q^{13} + q^{15} + 4q^{17} + 8q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 3161088
\( \Gamma_0(N) \)-optimal: yes
Manin constant: 1

Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 

\( L(E,1) \) ≈ \( 1.08133025901 \)

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(2\) \( I_11^{*} \) Additive -1 4 19 7
\(3\) \(1\) \( I_{7} \) Non-split multiplicative 1 1 7 7
\(7\) \(1\) \( I_{7} \) Non-split multiplicative 1 1 7 7
\(13\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1
\(41\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 
sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 

The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime Image of Galois representation
\(7\) B.6.1

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]
 

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 7.
Its isogeny class 179088.n consists of 2 curves linked by isogenies of degree 7.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 \(\Q(\sqrt{-1}) \) \(\Z/7\Z\) Not in database
3 3.1.89544.1 \(\Z/2\Z\) Not in database
6 \( x^{6} - 2 x^{5} - 24 x^{4} + 156 x^{3} + 69 x^{2} - 2218 x + 4714 \) \(\Z/14\Z\) Not in database
\( x^{6} - 86 x^{4} + 1849 x^{2} + 89544 \) \(\Z/2\Z \times \Z/2\Z\) 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.