Properties

Label 27225bm2
Conductor $27225$
Discriminant $-2.954\times 10^{17}$
j-invariant \( -121 \)
CM no
Rank $1$
Torsion structure trivial

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -68630, -27034378]) # or
 
sage: E = EllipticCurve("27225.g1")
 
gp: E = ellinit([1, -1, 1, -68630, -27034378]) \\ or
 
gp: E = ellinit("27225.g1")
 
magma: E := EllipticCurve([1, -1, 1, -68630, -27034378]); // or
 
magma: E := EllipticCurve("27225.g1");
 

\(y^2+xy+y=x^3-x^2-68630x-27034378\)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(\frac{231642}{529}, \frac{60237410}{12167}\right) \)
\(\hat{h}(P)\) ≈  $11.657911859562392117424816675$

Integral points

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

None

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 27225 \)  =  \(3^{2} \cdot 5^{2} \cdot 11^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-295443477095765625 \)  =  \(-1 \cdot 3^{6} \cdot 5^{6} \cdot 11^{10} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -121 \)  =  \(-1 \cdot 11^{2}\)
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);
 
Analytic rank: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(11.657911859562392117424816675\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.12971014033343119055189207020\)
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: \( 2 \)  = \( 2\cdot1\cdot1 \)
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 27225.2.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);
 

\( q - q^{2} - q^{4} - 2q^{7} + 3q^{8} + q^{13} + 2q^{14} - q^{16} + 5q^{17} - 6q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 221760
\( \Gamma_0(N) \)-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);
 

\( L'(E,1) \) ≈ \( 3.0242987665972193019379596074033682254 \)

Local data

This elliptic curve is 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\((j)_{-}\)
\(3\) \(2\) \(I_0^{*}\) Additive -1 2 6 0
\(5\) \(1\) \(I_0^{*}\) Additive 1 2 6 0
\(11\) \(1\) \(II^{*}\) Additive -1 2 10 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 X3.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 3 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 0 & 1 \\ 3 & 1 \end{array}\right)$ 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
\(11\) B.10.4

$p$-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.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ordinary add add ordinary add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary
$\lambda$-invariant(s) ? - - 1 - 1 1 1 1 1 1 1 1 1,1 1
$\mu$-invariant(s) ? - - 0 - 0 0 0 0 0 0 0 0 0,0 0

An entry ? indicates that the invariants have not yet been computed.

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 \(d=\) 11.
Its isogeny class 27225bm consists of 2 curves linked by isogenies of degree 11.

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
$3$ 3.1.484.1 \(\Z/2\Z\) Not in database
$6$ 6.0.937024.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
$8$ 8.2.2421502441875.1 \(\Z/3\Z\) Not in database
$10$ 10.0.162778775259375.1 \(\Z/11\Z\) Not in database
$12$ Deg 12 \(\Z/4\Z\) 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.