↑

Properties

Label 1260.d4
Conductor $1260$
Discriminant $3704400$
j-invariant \( \frac{10788913152}{8575} \)
CM no
Rank $1$
Torsion structure \(\Z/{6}\Z\)

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -348, 2497])
 
gp: E = ellinit([0, 0, 0, -348, 2497])
 
magma: E := EllipticCurve([0, 0, 0, -348, 2497]);
 

Minimal equation

Minimal equation

Simplified equation

\(y^2=x^3-348x+2497\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-348xz^2+2497z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-348x+2497\) Copy content Toggle raw display (homogenize, minimize)

Mordell-Weil group structure

\(\Z \oplus \Z/{6}\Z\)

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(6, 25\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $1.0338168530395322339413882219$

Torsion generators

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

\( \left(26, 105\right) \) Copy content Toggle raw display

Integral points

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

\((-16,\pm 63)\), \((-9,\pm 70)\), \((6,\pm 25)\), \((8,\pm 15)\), \( \left(11, 0\right) \), \((12,\pm 7)\), \((26,\pm 105)\), \((386,\pm 7575)\) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 1260 \)  =  $2^{2} \cdot 3^{2} \cdot 5 \cdot 7$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $3704400 $  =  $2^{4} \cdot 3^{3} \cdot 5^{2} \cdot 7^{3} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{10788913152}{8575} \)  =  $2^{14} \cdot 3^{3} \cdot 5^{-2} \cdot 7^{-3} \cdot 29^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.19074610089408388065165488009\dots$
Stable Faltings height: $-0.31495603145959197866956713629\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1.0338168530395322339413882219\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $2.4708965813577122815504004342\dots$
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: $ 36 $  = $ 3\cdot2\cdot2\cdot3 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $6$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $1$ (exact)
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);
 
Special value: $ L'(E,1) $ ≈ $ 2.5544545279253686398910944152 $

Modular invariants

Modular form   1260.2.a.d

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);
 

\( q - q^{5} + q^{7} - 4 q^{13} - 6 q^{17} + 2 q^{19} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 288
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1

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$(j)_{-}$
$2$ $3$ $IV$ Additive -1 2 4 0
$3$ $2$ $III$ Additive 1 2 3 0
$5$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$7$ $3$ $I_{3}$ Split multiplicative -1 1 3 3

Galois representations

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 $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 2.3.0.1
$3$ 3B.1.1 3.8.0.1

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
 

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type add add nonsplit split ss ord ord ord ord ss ord ord ord ord ss
$\lambda$-invariant(s) - - 1 2 1,1 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 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 $d=$ 2, 3 and 6.
Its isogeny class 1260.d consists of 4 curves linked by isogenies of degrees dividing 6.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{21}) \) \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$4$ 4.4.302400.2 \(\Z/12\Z\) Not in database
$6$ 6.0.21870000.2 \(\Z/3\Z \oplus \Z/6\Z\) Not in database
$8$ 8.8.4480842240000.2 \(\Z/2\Z \oplus \Z/12\Z\) Not in database
$8$ 8.0.8782450790400.9 \(\Z/2\Z \oplus \Z/12\Z\) Not in database
$9$ 9.3.67525382385720000.7 \(\Z/18\Z\) Not in database
$12$ Deg 12 \(\Z/6\Z \oplus \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/24\Z\) Not in database
$18$ 18.6.4691907907061498279186856033600000000.5 \(\Z/2\Z \oplus \Z/18\Z\) 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.