Properties

 Label 56a3 Conductor $56$ Discriminant $4917248$ j-invariant $$\frac{11090466}{2401}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

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Show commands: Magma / Oscar / PariGP / SageMath

Simplified equation

 $$y^2=x^3-59x-138$$ y^2=x^3-59x-138 (homogenize, simplify) $$y^2z=x^3-59xz^2-138z^3$$ y^2z=x^3-59xz^2-138z^3 (dehomogenize, simplify) $$y^2=x^3-59x-138$$ y^2=x^3-59x-138 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, 0, 0, -59, -138])

gp: E = ellinit([0, 0, 0, -59, -138])

magma: E := EllipticCurve([0, 0, 0, -59, -138]);

oscar: E = EllipticCurve([0, 0, 0, -59, -138])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

Mordell-Weil group structure

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

magma: MordellWeilGroup(E);

Torsion generators

$$\left(-6, 0\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

Integral points

$$\left(-6, 0\right)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

Invariants

 Conductor: $$56$$ = $2^{3} \cdot 7$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $4917248$ = $2^{11} \cdot 7^{4}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{11090466}{2401}$$ = $2 \cdot 3^{3} \cdot 7^{-4} \cdot 59^{3}$ 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: $-0.0020328267985874883465435369944\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-0.63741774231187068864567298166\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $1.1170620055320515\dots$ Szpiro ratio: $5.924007936828446\dots$

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: $1.7490966283766727306017206995\dots$ 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: $2$  = $1\cdot2$ 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: $2$ 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: $L(E,1)$ ≈ $0.87454831418833636530086034973$ 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

$\displaystyle 0.874548314 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.749097 \cdot 1.000000 \cdot 2}{2^2} \approx 0.874548314$

# 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("analytic 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

$$q + 2 q^{5} - q^{7} - 3 q^{9} - 4 q^{11} + 2 q^{13} - 6 q^{17} + 8 q^{19} + O(q^{20})$$

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: 8
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 not semistable. There are 2 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $1$ $II^{*}$ Additive -1 3 11 0
$7$ $2$ $I_{4}$ Non-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
$2$ 2B 8.24.0.58

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 = [[17, 8, 12, 33], [49, 8, 48, 9], [1, 0, 8, 1], [1, 4, 4, 17], [7, 6, 50, 51], [43, 36, 26, 13], [24, 1, 27, 14], [1, 8, 0, 1]]

GL(2,Integers(56)).subgroup(gens)

Gens := [[17, 8, 12, 33], [49, 8, 48, 9], [1, 0, 8, 1], [1, 4, 4, 17], [7, 6, 50, 51], [43, 36, 26, 13], [24, 1, 27, 14], [1, 8, 0, 1]];

sub<GL(2,Integers(56))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 56.48.0-56.v.1.8, level $$56 = 2^{3} \cdot 7$$, index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 17 & 8 \\ 12 & 33 \end{array}\right),\left(\begin{array}{rr} 49 & 8 \\ 48 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 50 & 51 \end{array}\right),\left(\begin{array}{rr} 43 & 36 \\ 26 & 13 \end{array}\right),\left(\begin{array}{rr} 24 & 1 \\ 27 & 14 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

The torsion field $K:=\Q(E[56])$ is a degree-$64512$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/56\Z)$.

Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 56a consists of 4 curves linked by isogenies of degrees dividing 4.

Twists

This elliptic curve is its own minimal quadratic twist.

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 $K$ $2$ $$\Q(\sqrt{2})$$ $$\Z/2\Z \oplus \Z/2\Z$$ 2.2.8.1-392.1-b3 $2$ $$\Q(\sqrt{-1})$$ $$\Z/4\Z$$ 2.0.4.1-392.1-b2 $2$ $$\Q(\sqrt{-2})$$ $$\Z/4\Z$$ 2.0.8.1-392.1-a2 $4$ $$\Q(\zeta_{8})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ 4.2.2048.1 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.16777216.2 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.9834496.1 $$\Z/8\Z$$ Not in database $8$ 8.0.10070523904.5 $$\Z/8\Z$$ Not in database $8$ 8.2.5377010688.2 $$\Z/6\Z$$ Not in database $16$ 16.0.24759631762948096.2 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ 16.0.1622647227216566419456.8 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ deg 16 $$\Z/12\Z$$ Not in database $16$ deg 16 $$\Z/12\Z$$ 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.

Iwasawa invariants

$p$ Reduction type 2 7 add nonsplit - 0 - 0

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

An entry - indicates that the invariants are not computed because the reduction is additive.

$p$-adic regulators

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