Properties

Label 3150br2
Conductor $3150$
Discriminant $-240045120000$
j-invariant \( -\frac{7620530425}{526848} \)
CM no
Rank $1$
Torsion structure \(\Z/{3}\Z\)

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Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -3155, 72947])
 
gp: E = ellinit([1, -1, 1, -3155, 72947])
 
magma: E := EllipticCurve([1, -1, 1, -3155, 72947]);
 

\(y^2+xy+y=x^3-x^2-3155x+72947\)  Toggle raw display

Mordell-Weil group structure

\(\Z\times \Z/{3}\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \(\left(39, 70\right)\)  Toggle raw display
\(\hat{h}(P)\) ≈  $0.13815541856997857598492782581$

Torsion generators

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

\( \left(19, 130\right) \)  Toggle raw display

Integral points

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

\( \left(-65, 46\right) \), \( \left(-65, 18\right) \), \( \left(-51, 340\right) \), \( \left(-51, -290\right) \), \( \left(-21, 370\right) \), \( \left(-21, -350\right) \), \( \left(3, 250\right) \), \( \left(3, -254\right) \), \( \left(19, 130\right) \), \( \left(19, -150\right) \), \( \left(33, 46\right) \), \( \left(33, -80\right) \), \( \left(39, 70\right) \), \( \left(39, -110\right) \), \( \left(49, 160\right) \), \( \left(49, -210\right) \), \( \left(75, 466\right) \), \( \left(75, -542\right) \), \( \left(159, 1810\right) \), \( \left(159, -1970\right) \), \( \left(579, 13570\right) \), \( \left(579, -14150\right) \), \( \left(669, 16900\right) \), \( \left(669, -17570\right) \), \( \left(118123, 40538530\right) \), \( \left(118123, -40656654\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 3150 \)  =  \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-240045120000 \)  =  \(-1 \cdot 2^{9} \cdot 3^{7} \cdot 5^{4} \cdot 7^{3} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{7620530425}{526848} \)  =  \(-1 \cdot 2^{-9} \cdot 3^{-1} \cdot 5^{2} \cdot 7^{-3} \cdot 673^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: \(0.93462356429511338019336642610\dots\)
Stable Faltings height: \(-0.15116188418364159037117597010\dots\)

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(0.13815541856997857598492782581\dots\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.97235150404184267083718384785\dots\)
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: \( 324 \)  = \( 3^{2}\cdot2^{2}\cdot3\cdot3 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(3\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form   3150.2.a.bg

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);
 

\( q + q^{2} + q^{4} + q^{7} + q^{8} - 6q^{11} - q^{13} + q^{14} + q^{16} - 3q^{17} - 4q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 5184
\( \Gamma_0(N) \)-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);
 

\( L'(E,1) \) ≈ \( 4.8360826453697636151715136329857148351 \)

Local data

This elliptic curve is not semistable. There are 4 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\((j)_{-}\)
\(2\) \(9\) \(I_{9}\) Split multiplicative -1 1 9 9
\(3\) \(4\) \(I_1^{*}\) Additive -1 2 7 1
\(5\) \(3\) \(IV\) Additive -1 2 4 0
\(7\) \(3\) \(I_{3}\) Split multiplicative -1 1 3 3

Galois representations

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

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.1.1

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 split add add split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss
$\lambda$-invariant(s) 2 - - 2 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 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.
Its isogeny class 3150br consists of 2 curves linked by isogenies of degree 3.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.1.4200.1 \(\Z/6\Z\) Not in database
$6$ 6.0.2963520000.1 \(\Z/2\Z \times \Z/6\Z\) Not in database
$6$ 6.0.110716875.2 \(\Z/3\Z \times \Z/3\Z\) Not in database
$9$ 9.3.26377102494421875.2 \(\Z/9\Z\) Not in database
$12$ Deg 12 \(\Z/12\Z\) Not in database
$18$ 18.0.41857146662060206427328000000000000.3 \(\Z/3\Z \times \Z/6\Z\) 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.