Properties

 Label 102b3 Conductor $102$ Discriminant $27060804$ j-invariant $$\frac{576615941610337}{27060804}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z \oplus \Z/{2}\Z$$

Related objects

Show commands: Magma / Oscar / PariGP / SageMath

Simplified equation

 $$y^2+xy=x^3-1734x-27936$$ y^2+xy=x^3-1734x-27936 (homogenize, simplify) $$y^2z+xyz=x^3-1734xz^2-27936z^3$$ y^2z+xyz=x^3-1734xz^2-27936z^3 (dehomogenize, simplify) $$y^2=x^3-2247291x-1296640170$$ y^2=x^3-2247291x-1296640170 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 0, 0, -1734, -27936])

gp: E = ellinit([1, 0, 0, -1734, -27936])

magma: E := EllipticCurve([1, 0, 0, -1734, -27936]);

oscar: E = EllipticCurve([1, 0, 0, -1734, -27936])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

Mordell-Weil group structure

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

magma: MordellWeilGroup(E);

Torsion generators

$$\left(-24, 12\right)$$, $$\left(48, -24\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

Integral points

$$\left(-24, 12\right)$$, $$\left(48, -24\right)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

Invariants

 Conductor: $$102$$ = $2 \cdot 3 \cdot 17$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $27060804$ = $2^{2} \cdot 3^{4} \cdot 17^{4}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{576615941610337}{27060804}$$ = $2^{-2} \cdot 3^{-4} \cdot 17^{-4} \cdot 83233^{3}$ comment: j-invariant  sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E);  oscar: j_invariant(E) Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.50050469536149959952921099595\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $0.50050469536149959952921099595\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E)

BSD invariants

 Analytic rank: $0$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $1$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $0.73983896389723910579048627192\dots$ comment: Real Period  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); Tamagawa product: $32$  = $2\cdot2^{2}\cdot2^{2}$ comment: Tamagawa numbers  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E);  oscar: tamagawa_numbers(E) Torsion order: $4$ comment: Torsion order  sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E));  oscar: prod(torsion_structure(E)[1]) Analytic order of Ш: $1$ (exact) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L(E,1)$ ≈ $1.4796779277944782115809725438$ comment: Special L-value  r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: [r,L1r] = ellanalyticrank(E); L1r/r!  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

BSD formula

$\displaystyle 1.479677928 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.739839 \cdot 1.000000 \cdot 32}{4^2} \approx 1.479677928$

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)

E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;

Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();

omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();

assert r == ar; print("anayltic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))

/* self-contained Magma code snippet for the BSD formula (checks rank, computes analyiic sha) */

E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;

sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);

reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);

assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);

Modular invariants

$$q + q^{2} + q^{3} + q^{4} - 2 q^{5} + q^{6} + q^{8} + q^{9} - 2 q^{10} - 4 q^{11} + q^{12} - 2 q^{13} - 2 q^{15} + q^{16} + q^{17} + q^{18} + 4 q^{19} + O(q^{20})$$

comment: q-expansion of modular form

sage: E.q_eigenform(20)

\\ actual modular form, use for small N

[mf,F] = mffromell(E)

Ser(mfcoefs(mf,20),q)

\\ or just the series

Ser(ellan(E,20),q)*q

magma: ModularForm(E);

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

Modular degree: 64
comment: Modular degree

sage: E.modular_degree()

gp: ellmoddegree(E)

magma: ModularDegree(E);

$\Gamma_0(N)$-optimal: no
Manin constant: 1
comment: Manin constant

magma: ManinConstant(E);

Local data

This elliptic curve is semistable. There are 3 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$3$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$17$ $4$ $I_{4}$ Split multiplicative -1 1 4 4

comment: Local data

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]

Galois representations

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
$2$ 2Cs 8.96.0.48

comment: mod p Galois image

sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

gens = [[1, 42, 0, 103], [1, 0, 8, 1], [3, 110, 94, 115], [105, 8, 12, 33], [129, 8, 128, 9], [5, 4, 132, 133], [1, 8, 0, 1]]

GL(2,Integers(136)).subgroup(gens)

Gens := [[1, 42, 0, 103], [1, 0, 8, 1], [3, 110, 94, 115], [105, 8, 12, 33], [129, 8, 128, 9], [5, 4, 132, 133], [1, 8, 0, 1]];

sub<GL(2,Integers(136))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$136 = 2^{3} \cdot 17$$, index $192$, genus $1$, and generators

$\left(\begin{array}{rr} 1 & 42 \\ 0 & 103 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 110 \\ 94 & 115 \end{array}\right),\left(\begin{array}{rr} 105 & 8 \\ 12 & 33 \end{array}\right),\left(\begin{array}{rr} 129 & 8 \\ 128 & 9 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 132 & 133 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.

The torsion field $K:=\Q(E[136])$ is a degree-$626688$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/136\Z)$.

$p$-adic regulators

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

Iwasawa invariants

$p$ Reduction type $\lambda$-invariant(s) 2 3 17 split split split 1 1 1 2 0 0

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

Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 102b consists of 6 curves linked by isogenies of degrees dividing 8.

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

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{2})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-2})$$ $$\Z/2\Z \oplus \Z/4\Z$$ 2.0.8.1-5202.5-i7 $2$ $$\Q(\sqrt{-1})$$ $$\Z/2\Z \oplus \Z/4\Z$$ 2.0.4.1-5202.2-e5 $4$ $$\Q(\zeta_{8})$$ $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(i, \sqrt{17})$$ $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $4$ 4.0.18432.1 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.4.28375309615104.5 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.0.5473632256.1 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $8$ 8.0.1358954496.9 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $8$ 8.2.236727913392.3 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ deg 16 $$\Z/8\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/12\Z$$ 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.