Properties

 Label 114b2 Conductor $114$ Discriminant $-42633378$ j-invariant $$-\frac{69173457625}{42633378}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Simplified equation

 $$y^2+xy=x^3+x^2-85x-473$$ y^2+xy=x^3+x^2-85x-473 (homogenize, simplify) $$y^2z+xyz=x^3+x^2z-85xz^2-473z^3$$ y^2z+xyz=x^3+x^2z-85xz^2-473z^3 (dehomogenize, simplify) $$y^2=x^3-110835x-20409138$$ y^2=x^3-110835x-20409138 (homogenize, minimize)

sage: E = EllipticCurve([1, 1, 0, -85, -473])

gp: E = ellinit([1, 1, 0, -85, -473])

magma: E := EllipticCurve([1, 1, 0, -85, -473]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

Mordell-Weil group structure

$$\Z/{2}\Z$$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(\frac{43}{4}, -\frac{43}{8}\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$114$$ = $2 \cdot 3 \cdot 19$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-42633378$ = $-1 \cdot 2 \cdot 3^{10} \cdot 19^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{69173457625}{42633378}$$ = $-1 \cdot 2^{-1} \cdot 3^{-10} \cdot 5^{3} \cdot 19^{-2} \cdot 821^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.16588295511894989169631062847\dots$ Stable Faltings height: $0.16588295511894989169631062847\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Real period: $0.76357205651559192806870953186\dots$ 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: $4$  = $1\cdot2\cdot2$ 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: $1$ (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: $L(E,1)$ ≈ $0.76357205651559192806870953186$

Modular invariants

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);

$$q - q^{2} - q^{3} + q^{4} + q^{6} + 4 q^{7} - q^{8} + q^{9} + 4 q^{11} - q^{12} - 4 q^{14} + q^{16} - 2 q^{17} - q^{18} + q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 40 $\Gamma_0(N)$-optimal: no Manin constant: 1

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$(j)_{-}$
$2$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$3$ $2$ $I_{10}$ Non-split multiplicative 1 1 10 10
$19$ $2$ $I_{2}$ Split multiplicative -1 1 2 2

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 8.6.0.5
sage: gens = [[1, 2, 2, 5], [305, 4, 154, 9], [1, 4, 0, 1], [97, 4, 194, 9], [172, 289, 57, 400], [453, 4, 452, 5], [2, 1, 227, 0], [1, 0, 4, 1], [3, 4, 8, 11]]

sage: GL(2,Integers(456)).subgroup(gens)

magma: Gens := [[1, 2, 2, 5], [305, 4, 154, 9], [1, 4, 0, 1], [97, 4, 194, 9], [172, 289, 57, 400], [453, 4, 452, 5], [2, 1, 227, 0], [1, 0, 4, 1], [3, 4, 8, 11]];

magma: sub<GL(2,Integers(456))|Gens>;

The image of the adelic Galois representation has level $456$, index $12$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 305 & 4 \\ 154 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 97 & 4 \\ 194 & 9 \end{array}\right),\left(\begin{array}{rr} 172 & 289 \\ 57 & 400 \end{array}\right),\left(\begin{array}{rr} 453 & 4 \\ 452 & 5 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 227 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$

$p$-adic regulators

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

All $p$-adic regulators are identically $1$ since the rank is $0$.

Iwasawa invariants

$p$ Reduction type $\lambda$-invariant(s) 2 3 19 nonsplit nonsplit split 0 0 1 1 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 114b 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:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-2})$$ $$\Z/2\Z \oplus \Z/2\Z$$ 2.0.8.1-6498.5-a3 $4$ 4.2.103968.2 $$\Z/4\Z$$ Not in database $8$ 8.0.13627293696.9 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.691798081536.14 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.2.369375586992.1 $$\Z/6\Z$$ Not in database $16$ deg 16 $$\Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ 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.