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Properties

Label 25410bj2
Conductor $25410$
Discriminant $9.113\times 10^{16}$
j-invariant \( \frac{135487869158881}{51438240000} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z \oplus \Z/{2}\Z\)

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Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy+y=x^3-129473x-10527244\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz+yz^2=x^3-129473xz^2-10527244z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-167796387x-490655695266\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, 0, 1, -129473, -10527244])
 
gp: E = ellinit([1, 0, 1, -129473, -10527244])
 
magma: E := EllipticCurve([1, 0, 1, -129473, -10527244]);
 
oscar: E = elliptic_curve([1, 0, 1, -129473, -10527244])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)

magma: MordellWeilGroup(E);
 

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$(-165, 2602)$$0.65058285753234229392480644639$$\infty$
$(-309, 154)$$0$$2$
$(395, -198)$$0$$2$

Integral points

\( \left(-309, 154\right) \), \( \left(-298, 1419\right) \), \( \left(-298, -1122\right) \), \( \left(-210, 2827\right) \), \( \left(-210, -2618\right) \), \( \left(-165, 2602\right) \), \( \left(-165, -2438\right) \), \( \left(395, -198\right) \), \( \left(420, 2827\right) \), \( \left(420, -3248\right) \), \( \left(780, 18667\right) \), \( \left(780, -19448\right) \), \( \left(1275, 42922\right) \), \( \left(1275, -44198\right) \), \( \left(13947, 1639594\right) \), \( \left(13947, -1653542\right) \) Copy content Toggle raw display

comment: Integral points
 
sage: E.integral_points()
 
magma: IntegralPoints(E);
 

Invariants

Conductor: $N$  =  \( 25410 \) = $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2}$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $\Delta$  =  $91125979892640000$ = $2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \cdot 11^{6} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: $j$  =  \( \frac{135487869158881}{51438240000} \) = $2^{-8} \cdot 3^{-8} \cdot 5^{-4} \cdot 7^{-2} \cdot 51361^{3}$
comment: j-invariant
 
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
oscar: j_invariant(E)
 
Endomorphism ring: $\mathrm{End}(E)$ = $\Z$
Geometric endomorphism ring: $\mathrm{End}(E_{\overline{\Q}})$  =  \(\Z\)    (no potential complex multiplication)
sage: E.has_cm()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{ST}(E)$ = $\mathrm{SU}(2)$
Faltings height: $h_{\mathrm{Faltings}}$ ≈ $1.9536562824548303548581618542$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $h_{\mathrm{stable}}$ ≈ $0.75470864605564508282719006522$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
$abc$ quality: $Q$ ≈ $1.0190986885579905$
Szpiro ratio: $\sigma_{m}$ ≈ $4.626614061331551$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$ = $ 1$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Mordell-Weil rank: $r$ = $ 1$
comment: Rank
 
sage: E.rank()
 
gp: [lower,upper] = ellrank(E)
 
magma: Rank(E);
 
Regulator: $\mathrm{Reg}(E/\Q)$ ≈ $0.65058285753234229392480644639$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $\Omega$ ≈ $0.25984421006050586052987717704$
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: $\prod_{p}c_p$ = $ 512 $  = $ 2\cdot2^{3}\cdot2^{2}\cdot2\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: $\#E(\Q)_{\mathrm{tor}}$ = $4$
comment: Torsion order
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
oscar: prod(torsion_structure(E)[1])
 
Special value: $ L'(E,1)$ ≈ $5.4096060382207394705747195760 $
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);
 
Analytic order of Ш: Ш${}_{\mathrm{an}}$  ≈  $1$    (rounded)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 

BSD formula

$\displaystyle 5.409606038 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.259844 \cdot 0.650583 \cdot 512}{4^2} \approx 5.409606038$

# 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 analytic 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   25410.2.a.bl

\( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + q^{7} - q^{8} + q^{9} - q^{10} + q^{12} + 2 q^{13} - q^{14} + q^{15} + q^{16} - 2 q^{17} - q^{18} - 4 q^{19} + O(q^{20}) \) 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: 327680
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 at primes of bad reduction

This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $\mathrm{ord}_p(N)$ $\mathrm{ord}_p(\Delta)$ $\mathrm{ord}_p(\mathrm{den}(j))$
$2$ $2$ $I_{8}$ nonsplit multiplicative 1 1 8 8
$3$ $8$ $I_{8}$ split multiplicative -1 1 8 8
$5$ $4$ $I_{4}$ split multiplicative -1 1 4 4
$7$ $2$ $I_{2}$ split multiplicative -1 1 2 2
$11$ $4$ $I_0^{*}$ additive -1 2 6 0

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 8.48.0.98

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, 16, 0, 1], [18465, 16, 18464, 17], [5281, 15136, 16566, 16897], [1695, 15136, 16874, 13729], [13113, 15136, 6116, 11881], [1, 0, 16, 1], [6161, 16808, 0, 1], [1, 16, 4, 65], [6733, 16808, 7040, 617], [6719, 0, 0, 18479]]
 
GL(2,Integers(18480)).subgroup(gens)
 
Gens := [[1, 16, 0, 1], [18465, 16, 18464, 17], [5281, 15136, 16566, 16897], [1695, 15136, 16874, 13729], [13113, 15136, 6116, 11881], [1, 0, 16, 1], [6161, 16808, 0, 1], [1, 16, 4, 65], [6733, 16808, 7040, 617], [6719, 0, 0, 18479]];
 
sub<GL(2,Integers(18480))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 18480 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $768$, genus $13$, and generators

$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 18465 & 16 \\ 18464 & 17 \end{array}\right),\left(\begin{array}{rr} 5281 & 15136 \\ 16566 & 16897 \end{array}\right),\left(\begin{array}{rr} 1695 & 15136 \\ 16874 & 13729 \end{array}\right),\left(\begin{array}{rr} 13113 & 15136 \\ 6116 & 11881 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 6161 & 16808 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 4 & 65 \end{array}\right),\left(\begin{array}{rr} 6733 & 16808 \\ 7040 & 617 \end{array}\right),\left(\begin{array}{rr} 6719 & 0 \\ 0 & 18479 \end{array}\right)$.

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

The torsion field $K:=\Q(E[18480])$ is a degree-$19619905536000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/18480\Z)$.

The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.

$\ell$ Reduction type Serre weight Serre conductor
$2$ nonsplit multiplicative $4$ \( 121 = 11^{2} \)
$3$ split multiplicative $4$ \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
$5$ split multiplicative $6$ \( 5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2} \)
$7$ split multiplicative $8$ \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \)
$11$ additive $62$ \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4 and 8.
Its isogeny class 25410bj consists of 8 curves linked by isogenies of degrees dividing 16.

Twists

The minimal quadratic twist of this elliptic curve is 210e2, its twist by $-11$.

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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-11}) \) \(\Z/2\Z \oplus \Z/8\Z\) 2.0.11.1-44100.5-r4
$4$ \(\Q(\sqrt{-7}, \sqrt{11})\) \(\Z/2\Z \oplus \Z/4\Z\) not in database
$4$ \(\Q(\sqrt{7}, \sqrt{11})\) \(\Z/2\Z \oplus \Z/4\Z\) not in database
$8$ 8.0.8999178496.1 \(\Z/4\Z \oplus \Z/8\Z\) not in database
$8$ 8.0.7289334581760000.272 \(\Z/2\Z \oplus \Z/16\Z\) not in database
$8$ 8.0.3035957760000.5 \(\Z/2\Z \oplus \Z/16\Z\) not in database
$8$ deg 8 \(\Z/2\Z \oplus \Z/6\Z\) not in database
$16$ deg 16 \(\Z/2\Z \oplus \Z/8\Z\) not in database
$16$ deg 16 \(\Z/2\Z \oplus \Z/8\Z\) not in database
$16$ deg 16 \(\Z/2\Z \oplus \Z/24\Z\) 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 nonsplit split split split add ord ord ord ord ord ss ord ord ord ss
$\lambda$-invariant(s) 4 2 2 2 - 3 1 3 1 1 1,1 1 1 1 1,1
$\mu$-invariant(s) 0 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

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.