Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-1557845383x-23653796228447\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-1557845383xz^2-23653796228447z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-2018967617043x-1103561232320170962\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(-\frac{92917}{4}, \frac{92917}{8}\right) \)
Integral points
None
Invariants
Conductor: | \( 35490 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $273635311017579486321904740 $ | = | $2^{2} \cdot 3^{8} \cdot 5 \cdot 7^{16} \cdot 13^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{86623684689189325642735681}{56690726941459561860} \) | = | $2^{-2} \cdot 3^{-8} \cdot 5^{-1} \cdot 7^{-16} \cdot 13^{-1} \cdot 23^{3} \cdot 4007^{3} \cdot 4801^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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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: | $4.0124729709612045816723914786\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.7299982922304362136456477578\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0354092118359115\dots$ | |||
Szpiro ratio: | $7.169348751826162\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.024031757182921275669038658070\dots$ | 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);
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Tamagawa product: | $ 128 $ = $ 2\cdot2\cdot1\cdot2^{4}\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)
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Torsion order: | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Analytic order of Ш: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 0.76901622985348082140923705825 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 0.769016230 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.024032 \cdot 1.000000 \cdot 128}{2^2} \approx 0.769016230$
Modular invariants
Modular form 35490.2.a.f
For more coefficients, see the Downloads section to the right.
Modular degree: | 22020096 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$3$ | $2$ | $I_{8}$ | Non-split multiplicative | 1 | 1 | 8 | 8 |
$5$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$7$ | $16$ | $I_{16}$ | Split multiplicative | -1 | 1 | 16 | 16 |
$13$ | $2$ | $I_{1}^{*}$ | Additive | 1 | 2 | 7 | 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 | 16.48.0.121 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 14560 = 2^{5} \cdot 5 \cdot 7 \cdot 13 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 4161 & 32 \\ 8336 & 513 \end{array}\right),\left(\begin{array}{rr} 12741 & 32 \\ 12756 & 513 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 11998 & 12555 \end{array}\right),\left(\begin{array}{rr} 2934 & 3 \\ 13005 & 14348 \end{array}\right),\left(\begin{array}{rr} 8191 & 32 \\ 4550 & 1 \end{array}\right),\left(\begin{array}{rr} 14529 & 32 \\ 14528 & 33 \end{array}\right),\left(\begin{array}{rr} 5584 & 14535 \\ 13641 & 11374 \end{array}\right)$.
The torsion field $K:=\Q(E[14560])$ is a degree-$12984810209280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/14560\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 8 and 16.
Its isogeny class 35490.f
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 2730.w1, its twist by $13$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{65}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-5}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-13}) \) | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-5}, \sqrt{-13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{2}, \sqrt{-13})\) | \(\Z/8\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-10}, \sqrt{-13})\) | \(\Z/8\Z\) | Not in database |
$8$ | 8.4.308915776000000.14 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.19307236000000.34 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.1169858560000.4 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.0.1265319018496000000.6 | \(\Z/16\Z\) | Not in database |
$8$ | 8.0.1265319018496000000.9 | \(\Z/16\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | \(\Z/16\Z\) | Not in database |
$16$ | 16.0.1601032218567680790102016000000000000.32 | \(\Z/2\Z \oplus \Z/16\Z\) | 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 | 7 | 13 |
---|---|---|---|---|---|
Reduction type | nonsplit | nonsplit | nonsplit | split | add |
$\lambda$-invariant(s) | 4 | 0 | 0 | 5 | - |
$\mu$-invariant(s) | 3 | 0 | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ 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 $0$.