Properties

Label 1296.g1
Conductor $1296$
Discriminant $-278628139008$
j-invariant \( -\frac{189613868625}{128} \)
CM no
Rank $0$
Torsion structure trivial

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -155115, 23514138])
 
gp: E = ellinit([0, 0, 0, -155115, 23514138])
 
magma: E := EllipticCurve([0, 0, 0, -155115, 23514138]);
 

\(y^2=x^3-155115x+23514138\)  Toggle raw display

Mordell-Weil group structure

trivial

Integral points

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

\(\)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 1296 \)  =  \(2^{4} \cdot 3^{4}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-278628139008 \)  =  \(-1 \cdot 2^{19} \cdot 3^{12} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{189613868625}{128} \)  =  \(-1 \cdot 2^{-7} \cdot 3^{3} \cdot 5^{3} \cdot 383^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: \(1.5118214037716403369456949906\dots\)
Stable Faltings height: \(-0.27993806545641466386678236778\dots\)

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: \(0.80848602428825605028754866728\dots\)
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: \( 2 \)  = \( 2\cdot1 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(1\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form   1296.2.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);
 

\( q - 2q^{7} + 3q^{11} + 2q^{13} - 3q^{17} + q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 3024
\( \Gamma_0(N) \)-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);
 

\( L(E,1) \) ≈ \( 1.6169720485765121005750973345518439593 \)

Local data

This elliptic curve is not semistable. There are 2 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\((j)_{-}\)
\(2\) \(2\) \(I_{11}^{*}\) Additive -1 4 19 7
\(3\) \(1\) \(II^{*}\) Additive 1 4 12 0

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 X4.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 7 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 1 & 1 \end{array}\right)$ and has index 2.

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
\(3\) B
\(7\) B

$p$-adic data

$p$-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 11 13 17 19 23 29 31 37 41 43 47
Reduction type add add ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) - - 0,0 0 0 0 0 2 0 0 0 0 0 2 0
$\mu$-invariant(s) - - 0,0 0 0 0 0 0 0 0 0 0 0 0 0

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 \(d=\) 3, 7 and 21.
Its isogeny class 1296.g consists of 4 curves linked by isogenies of degrees dividing 21.

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

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{3}) \) \(\Z/3\Z\) 2.2.12.1-1458.1-i1
$3$ 3.1.648.1 \(\Z/2\Z\) Not in database
$6$ 6.0.3359232.4 \(\Z/2\Z \times \Z/2\Z\) Not in database
$6$ 6.0.186624.1 \(\Z/3\Z\) Not in database
$6$ 6.6.7057326528.2 \(\Z/7\Z\) Not in database
$6$ 6.2.20155392.5 \(\Z/6\Z\) Not in database
$12$ 12.2.5777633090469888.3 \(\Z/4\Z\) Not in database
$12$ 12.0.313456656384.1 \(\Z/3\Z \times \Z/3\Z\) Not in database
$12$ 12.0.6499837226778624.48 \(\Z/2\Z \times \Z/6\Z\) Not in database
$12$ 12.12.448252719505312813056.1 \(\Z/21\Z\) Not in database
$18$ 18.6.1376809511370776442839236608.2 \(\Z/9\Z\) Not in database
$18$ 18.0.3537182715531733726396416.1 \(\Z/6\Z\) Not in database
$18$ 18.6.1439728439119866152968380211003392.1 \(\Z/14\Z\) Not in database

We only show fields where the torsion growth is primitive.