Properties

Label 47775.be1
Conductor $47775$
Discriminant $-9.537\times 10^{22}$
j-invariant \( \frac{48412981936758748562855}{77853743274432041397} \)
CM no
Rank $0$
Torsion structure trivial

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Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy=x^3+8124402x-11887136703\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3+8124402xz^2-11887136703z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3+10529224965x-554637837690090\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, 0, 0, 8124402, -11887136703])
 
gp: E = ellinit([1, 0, 0, 8124402, -11887136703])
 
magma: E := EllipticCurve([1, 0, 0, 8124402, -11887136703]);
 
oscar: E = EllipticCurve([1, 0, 0, 8124402, -11887136703])
 
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: \( 47775 \)  =  $3 \cdot 5^{2} \cdot 7^{2} \cdot 13$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $-95370835511179250711325 $  =  $-1 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13^{17} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( \frac{48412981936758748562855}{77853743274432041397} \)  =  $3^{-2} \cdot 5 \cdot 7 \cdot 13^{-17} \cdot 19^{3} \cdot 41^{3} \cdot 14303^{3}$
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: $3.0944938779831080615044528905\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $2.5019358677348724482201008777\dots$
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: $0.056323104404563911796967313729\dots$
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: $ 34 $  = $ 2\cdot1\cdot1\cdot17 $
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) $ ≈ $ 1.9149855497551730010968886668 $
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

$\displaystyle 1.914985550 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.056323 \cdot 1.000000 \cdot 34}{1^2} \approx 1.914985550$

# 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

Modular form 47775.2.a.be

\( q - q^{2} + q^{3} - q^{4} - q^{6} + 3 q^{8} + q^{9} - q^{12} + q^{13} - q^{16} + 2 q^{17} - q^{18} - q^{19} + O(q^{20}) \) Copy content Toggle raw display

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: 3818880
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data

This elliptic curve is not semistable. There are 4 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
$5$ $1$ $II$ Additive 1 2 2 0
$7$ $1$ $II$ Additive -1 2 2 0
$13$ $17$ $I_{17}$ Split multiplicative -1 1 17 17

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 $\ell$.

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 = [[41, 2, 41, 3], [1, 2, 0, 1], [1, 0, 2, 1], [27, 2, 27, 3], [1, 1, 51, 0], [51, 2, 50, 3]]
 
GL(2,Integers(52)).subgroup(gens)
 
Gens := [[41, 2, 41, 3], [1, 2, 0, 1], [1, 0, 2, 1], [27, 2, 27, 3], [1, 1, 51, 0], [51, 2, 50, 3]];
 
sub<GL(2,Integers(52))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 52.2.0.a.1, level \( 52 = 2^{2} \cdot 13 \), index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 41 & 2 \\ 41 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 27 & 2 \\ 27 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 51 & 0 \end{array}\right),\left(\begin{array}{rr} 51 & 2 \\ 50 & 3 \end{array}\right)$.

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

The torsion field $K:=\Q(E[52])$ is a degree-$1257984$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/52\Z)$.

Isogenies

gp: ellisomat(E)
 

This curve has no rational isogenies. Its isogeny class 47775.be consists of this curve only.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.1.63700.1 \(\Z/2\Z\) Not in database
$6$ 6.0.210999880000.1 \(\Z/2\Z \oplus \Z/2\Z\) 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.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ord split add add ss split ord ord ord ord ord ord ss ord ord
$\lambda$-invariant(s) 2 1 - - 0,0 1 2 0 0 0 0 0 0,0 0 0
$\mu$-invariant(s) 0 0 - - 0,0 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.

$p$-adic regulators

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

Additional information

The elliptic curves 47775.be1 and 3675.g1 are $17$-congruent, meaning that their 17-torsion subgroups are isomorphic as Galois modules. The isomorphism in this case is antisymplectic, and this is the only known pair of non-isogenous elliptic curves (up to simultaneous quadratic twist) which are antisymplectically 17-congruent.

There are no known examples of congruences modulo primes greater than 17.

See N. Billerey, "On some remarkable congruences between two eliptic curves" [arXiv:1605.09205]; T. Fisher, "On families of 7-congruent and 11-congruent elliptic curves" [MR:3356045, 10.1112/S1461157014000059]; N. Freitas and A. Kraus, "On symplectic isomorphisms of the $p$-torsion of elliptic curves", [arXiv:1607.01218]. See also T. Fisher, "On pairs of 17-congruent elliptic curves" [arXiv:2106.02033] for an example of a symplectic mod-17 congruence.