Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-271202x-54271924\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-271202xz^2-54271924z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-4339227x-3477742346\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(-\frac{1221}{4}, \frac{1217}{8}\right) \)
Integral points
None
Invariants
Conductor: | \( 1035 \) | = | $3^{2} \cdot 5 \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $994721923828125 $ | = | $3^{11} \cdot 5^{12} \cdot 23 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{3026030815665395929}{1364501953125} \) | = | $3^{-5} \cdot 5^{-12} \cdot 23^{-1} \cdot 1446409^{3}$ | 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: | $1.8355800281759740762542254904\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $1.2862738838419192305566028719\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $1.0132386828274675\dots$ | |||
Szpiro ratio: | $7.079277995049867\dots$ |
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
magma: Regulator(E);
|
Real period: | $0.20921278035518820690275772722\dots$ | 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: | $ 24 $ = $ 2\cdot( 2^{2} \cdot 3 )\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: | $2$ | 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) $ ≈ $ 1.2552766821311292414165463633 $ | 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
$\displaystyle 1.255276682 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.209213 \cdot 1.000000 \cdot 24}{2^2} \approx 1.255276682$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 9600 | 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 $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $I_{5}^{*}$ | additive | -1 | 2 | 11 | 5 |
$5$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
$23$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 | 4.12.0.8 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 2419 & 2418 \\ 1738 & 355 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1657 & 8 \\ 1108 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 1328 & 3 \\ 485 & 2 \end{array}\right),\left(\begin{array}{rr} 916 & 2759 \\ 1817 & 2754 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 2754 & 2755 \end{array}\right),\left(\begin{array}{rr} 2419 & 2416 \\ 1058 & 2421 \end{array}\right),\left(\begin{array}{rr} 2753 & 8 \\ 2752 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[2760])$ is a degree-$196977623040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2760\Z)$.
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 |
---|---|---|---|
$2$ | good | $2$ | \( 207 = 3^{2} \cdot 23 \) |
$3$ | additive | $8$ | \( 23 \) |
$5$ | split multiplicative | $6$ | \( 207 = 3^{2} \cdot 23 \) |
$23$ | split multiplicative | $24$ | \( 45 = 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 1035.c
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 345.e1, its twist by $-3$.
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:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{69}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z\) | not in database |
$2$ | \(\Q(\sqrt{-69}) \) | \(\Z/4\Z\) | not in database |
$4$ | \(\Q(i, \sqrt{69})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$4$ | 4.2.5256144.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$8$ | 8.0.442032795979776.14 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
$8$ | 8.0.15795855360000.37 | \(\Z/8\Z\) | not in database |
$8$ | deg 8 | \(\Z/8\Z\) | not in database |
$8$ | 8.2.49572993627.2 | \(\Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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
$p$ | 2 | 3 | 5 | 23 |
---|---|---|---|---|
Reduction type | ord | add | split | split |
$\lambda$-invariant(s) | 3 | - | 1 | 1 |
$\mu$-invariant(s) | 1 | - | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ 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 $0$.