Properties

 Label 93600.b1 Conductor $93600$ Discriminant $-1.516\times 10^{13}$ j-invariant $$-\frac{2963520}{13}$$ CM no Rank $2$ Torsion structure trivial

Related objects

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -31500, 2160000])

gp: E = ellinit([0, 0, 0, -31500, 2160000])

magma: E := EllipticCurve([0, 0, 0, -31500, 2160000]);

$$y^2=x^3-31500x+2160000$$

Mordell-Weil group structure

$\Z^2$

Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(150, 900\right)$$ $$\left(-50, 1900\right)$$ $\hat{h}(P)$ ≈ $0.30135539068134003916379916009$ $1.9237662382858545480136688264$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-75,\pm 2025)$$, $$(-50,\pm 1900)$$, $$(6,\pm 1404)$$, $$(94,\pm 172)$$, $$(96,\pm 144)$$, $$(100,\pm 100)$$, $$(150,\pm 900)$$, $$(3525,\pm 209025)$$, $$(24774,\pm 3899268)$$

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: $-15163200000000$ = $-1 \cdot 2^{12} \cdot 3^{6} \cdot 5^{8} \cdot 13$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{2963520}{13}$$ = $-1 \cdot 2^{6} \cdot 3^{3} \cdot 5 \cdot 7^{3} \cdot 13^{-1}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.3821290464109346297986030084\dots$ Stable Faltings height: $-0.93328288677246577505009128700\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $2$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.57746648789936999422063815788\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.70363920504087738964418760728\dots$ 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: $24$  = $2^{2}\cdot2\cdot3\cdot1$ 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: $1$ (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: $L^{(2)}(E,1)/2!$ ≈ $9.7518734515982434887981005577575135933$

Modular invariants

Modular form 93600.2.a.b

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);

$$q - 5q^{7} - q^{11} + q^{13} + q^{17} - 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 307200 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $4$ $I_3^{*}$ Additive 1 5 12 0
$3$ $2$ $I_0^{*}$ Additive -1 2 6 0
$5$ $3$ $IV^{*}$ Additive -1 2 8 0
$13$ $1$ $I_{1}$ 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 $\ell$.

$p$-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

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 add add add ordinary ordinary split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary - - - 2 2 3 2 2 2 2 2 2 2 2 2 - - - 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 no rational isogenies. Its isogeny class 93600.b consists of this curve only.

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:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.1300.1 $$\Z/2\Z$$ Not in database $6$ 6.0.87880000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ Deg 8 $$\Z/3\Z$$ 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.