Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+y=x^3+x^2+3922x-12346\) | (homogenize, simplify) |
\(y^2z+yz^2=x^3+x^2z+3922xz^2-12346z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+5082480x-636994800\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(13, 202\right)\) | \(\left(823, 23692\right)\) |
$\hat{h}(P)$ | ≈ | $0.19674605507251634496854074080$ | $0.70655887919400420419955734153$ |
Integral points
\( \left(4, 58\right) \), \( \left(4, -59\right) \), \( \left(7, 124\right) \), \( \left(7, -125\right) \), \( \left(13, 202\right) \), \( \left(13, -203\right) \), \( \left(43, 487\right) \), \( \left(43, -488\right) \), \( \left(73, 817\right) \), \( \left(73, -818\right) \), \( \left(94, 1093\right) \), \( \left(94, -1094\right) \), \( \left(148, 1957\right) \), \( \left(148, -1958\right) \), \( \left(202, 3010\right) \), \( \left(202, -3011\right) \), \( \left(238, 3802\right) \), \( \left(238, -3803\right) \), \( \left(472, 10354\right) \), \( \left(472, -10355\right) \), \( \left(823, 23692\right) \), \( \left(823, -23693\right) \), \( \left(1147, 38920\right) \), \( \left(1147, -38921\right) \), \( \left(3709, 225946\right) \), \( \left(3709, -225947\right) \), \( \left(8113, 730822\right) \), \( \left(8113, -730823\right) \)
Invariants
Conductor: | \( 12675 \) | = | $3 \cdot 5^{2} \cdot 13^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-3940568584875 $ | = | $-1 \cdot 3^{15} \cdot 5^{3} \cdot 13^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{24288219136}{14348907} \) | = | $2^{12} \cdot 3^{-15} \cdot 181^{3}$ | 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: | $1.1062225595284401952000151994\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $0.062625742054530917536453505702\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $0.13815271348697421273659224976\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.45905243613443802861890148952\dots$ | 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: | $ 60 $ = $ ( 3 \cdot 5 )\cdot2\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: | $1$ | 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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
Special value: | $ L^{(2)}(E,1)/2! $ ≈ $ 3.8051603810867127045273381357 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 3.805160381 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.459052 \cdot 0.138153 \cdot 60}{1^2} \approx 3.805160381$
Modular invariants
Modular form 12675.2.a.e
For more coefficients, see the Downloads section to the right.
Modular degree: | 37440 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $15$ | $I_{15}$ | Split multiplicative | -1 | 1 | 15 | 15 |
$5$ | $2$ | $III$ | Additive | -1 | 2 | 3 | 0 |
$13$ | $2$ | $III$ | Additive | -1 | 2 | 3 | 0 |
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 |
---|---|---|
$3$ | 3Nn | 3.3.0.1 |
$5$ | 5B | 5.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \), index $288$, genus $9$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 30 & 1 \end{array}\right),\left(\begin{array}{rr} 91 & 30 \\ 315 & 187 \end{array}\right),\left(\begin{array}{rr} 1 & 30 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 361 & 30 \\ 360 & 31 \end{array}\right),\left(\begin{array}{rr} 47 & 10 \\ 25 & 103 \end{array}\right),\left(\begin{array}{rr} 6 & 11 \\ 175 & 321 \end{array}\right),\left(\begin{array}{rr} 131 & 30 \\ 15 & 61 \end{array}\right)$.
The torsion field $K:=\Q(E[390])$ is a degree-$12579840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/390\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 12675.e
consists of 2 curves linked by isogenies of
degree 5.
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 |
---|---|---|---|
$3$ | 3.1.780.1 | \(\Z/2\Z\) | Not in database |
$4$ | 4.0.2197.1 | \(\Z/5\Z\) | Not in database |
$6$ | 6.0.118638000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$8$ | 8.2.2036310046875.1 | \(\Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | deg 12 | \(\Z/10\Z\) | Not in database |
$16$ | deg 16 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$16$ | deg 16 | \(\Z/15\Z\) | Not in database |
$20$ | 20.4.238352888329377197660505771636962890625.2 | \(\Z/5\Z\) | 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.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | ss | split | add | ord | ord | add | ord | ord | ord | ss | ss | ord | ord | ord | ord |
$\lambda$-invariant(s) | 2,3 | 3 | - | 2 | 2 | - | 2 | 2 | 2 | 2,2 | 2,2 | 2 | 2 | 2 | 2 |
$\mu$-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.