Properties

 Label 57.b1 Conductor $57$ Discriminant $-22284891$ j-invariant $$-\frac{9358714467168256}{22284891}$$ CM no Rank $0$ Torsion structure trivial

Related objects

Show commands: Magma / Oscar / PariGP / SageMath

Simplified equation

 $$y^2+y=x^3+x^2-4390x-113432$$ y^2+y=x^3+x^2-4390x-113432 (homogenize, simplify) $$y^2z+yz^2=x^3+x^2z-4390xz^2-113432z^3$$ y^2z+yz^2=x^3+x^2z-4390xz^2-113432z^3 (dehomogenize, simplify) $$y^2=x^3-5689872x-5223994992$$ y^2=x^3-5689872x-5223994992 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, 1, 1, -4390, -113432])

gp: E = ellinit([0, 1, 1, -4390, -113432])

magma: E := EllipticCurve([0, 1, 1, -4390, -113432]);

oscar: E = EllipticCurve([0, 1, 1, -4390, -113432])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

Mordell-Weil group structure

trivial

magma: MordellWeilGroup(E);

Integral points

None

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

Invariants

 Conductor: $$57$$ = $3 \cdot 19$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $-22284891$ = $-1 \cdot 3^{2} \cdot 19^{5}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$-\frac{9358714467168256}{22284891}$$ = $-1 \cdot 2^{12} \cdot 3^{-2} \cdot 19^{-5} \cdot 13171^{3}$ 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: $0.65140725122842848756374343035\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $0.65140725122842848756374343035\dots$ 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: $0.29325479582828399279210747717\dots$ 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: $2$  = $2\cdot1$ 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: $1$ 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: $L(E,1)$ ≈ $0.58650959165656798558421495434$ 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

$\displaystyle 0.586509592 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.293255 \cdot 1.000000 \cdot 2}{1^2} \approx 0.586509592$

# 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

$$q - 2 q^{2} + q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + 3 q^{7} + q^{9} - 2 q^{10} - 3 q^{11} + 2 q^{12} - 6 q^{13} - 6 q^{14} + q^{15} - 4 q^{16} + 3 q^{17} - 2 q^{18} - q^{19} + O(q^{20})$$

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);

Modular degree: 60
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 2 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$3$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$19$ $1$ $I_{5}$ Non-split multiplicative 1 1 5 5

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
$5$ 5B.1.2 5.24.0.3

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, 0, 10, 1], [21, 10, 105, 51], [1, 10, 0, 1], [6, 13, 135, 71], [181, 10, 180, 11], [7, 10, 150, 171]]

GL(2,Integers(190)).subgroup(gens)

Gens := [[1, 0, 10, 1], [21, 10, 105, 51], [1, 10, 0, 1], [6, 13, 135, 71], [181, 10, 180, 11], [7, 10, 150, 171]];

sub<GL(2,Integers(190))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$190 = 2 \cdot 5 \cdot 19$$, index $48$, genus $1$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 10 \\ 105 & 51 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 135 & 71 \end{array}\right),\left(\begin{array}{rr} 181 & 10 \\ 180 & 11 \end{array}\right),\left(\begin{array}{rr} 7 & 10 \\ 150 & 171 \end{array}\right)$.

The torsion field $K:=\Q(E[190])$ is a degree-$13689$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/190\Z)$.

$p$-adic regulators

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

Iwasawa invariants

$p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 19 ss split ord nonsplit 1,4 3 2 0 0,0 0 1 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.

Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 5.
Its isogeny class 57.b consists of 2 curves linked by isogenies of degree 5.

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 $K$ $3$ 3.1.76.1 $$\Z/2\Z$$ Not in database $4$ $$\Q(\zeta_{5})$$ $$\Z/5\Z$$ Not in database $5$ 5.1.253125.1 $$\Z/5\Z$$ Not in database $6$ 6.0.109744.2 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $8$ 8.2.23085974187.2 $$\Z/3\Z$$ Not in database $12$ deg 12 $$\Z/4\Z$$ Not in database $12$ deg 12 $$\Z/10\Z$$ Not in database $15$ 15.1.41121891520593750000000000.1 $$\Z/10\Z$$ Not in database $20$ 20.0.513156902790069580078125.1 $$\Z/5\Z \oplus \Z/5\Z$$ 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.