Properties

Label 93600.bf2
Conductor $93600$
Discriminant $-4.442\times 10^{14}$
j-invariant \( -\frac{601211584}{609375} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z\)

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -15825, -1271000])
 
gp: E = ellinit([0, 0, 0, -15825, -1271000])
 
magma: E := EllipticCurve([0, 0, 0, -15825, -1271000]);
 

\(y^2=x^3-15825x-1271000\)  Toggle raw display

Mordell-Weil group structure

$\Z^2 \times \Z/{2}\Z$

Infinite order Mordell-Weil generators and heights

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

$P$ =  \(\left(965, 29700\right)\)  Toggle raw display\(\left(315, 5000\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $3.8974918429867082982683019528$$4.0259747657982165689888856977$

Torsion generators

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

\( \left(155, 0\right) \)  Toggle raw display

Integral points

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

\( \left(155, 0\right) \), \((164,\pm 738)\), \((315,\pm 5000)\), \((965,\pm 29700)\), \((6405,\pm 512500)\)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 93600 \)  =  $2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 13$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-444234375000000 $  =  $-1 \cdot 2^{6} \cdot 3^{7} \cdot 5^{12} \cdot 13 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{601211584}{609375} \)  =  $-1 \cdot 2^{6} \cdot 3^{-1} \cdot 5^{-6} \cdot 13^{-1} \cdot 211^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.5061819512377511439620497207\dots$
Stable Faltings height: $-0.19441673959332654374456862510\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $2$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $12.599935432529499534864444077\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.20449985079632596610236995723\dots$
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: $ 16 $  = $ 2\cdot2\cdot2^{2}\cdot1 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $2$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $1$ (rounded)
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^{(2)}(E,1)/2! $ ≈ $ 10.306739663982494126365539904408911712 $

Modular invariants

Modular form 93600.2.a.bf

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 368640
$ \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$(j)_{-}$
$2$ $2$ $III$ Additive 1 5 6 0
$3$ $2$ $I_1^{*}$ Additive -1 2 7 1
$5$ $4$ $I_6^{*}$ Additive 1 2 12 6
$13$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

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 $\GL(2,\Z_\ell)$ 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

$p$-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 add ordinary ordinary nonsplit ss ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) - - - 2 2 2 2,4 2 2 4 2,2 2 2 2 2
$\mu$-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 $d=$ 2.
Its isogeny class 93600.bf consists of 2 curves linked by isogenies of degree 2.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-39}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$4$ 4.2.249600.3 \(\Z/4\Z\) Not in database
$8$ 8.0.2251996007040000.69 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.0.94758543360000.4 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/6\Z\) 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.