Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-15843x-1441118\) | (homogenize, simplify) |
\(y^2z=x^3-15843xz^2-1441118z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-15843x-1441118\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
$P$ | $\hat{h}(P)$ | Order |
---|---|---|
$(279, 3982)$ | $5.2012513759725045021929325719$ | $\infty$ |
$(158, 0)$ | $0$ | $2$ |
Integral points
\( \left(158, 0\right) \), \((279,\pm 3982)\)
Invariants
Conductor: | $N$ | = | \( 720 \) | = | $2^{4} \cdot 3^{2} \cdot 5$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $\Delta$ | = | $-642684100792320$ | = | $-1 \cdot 2^{12} \cdot 3^{22} \cdot 5 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | $j$ | = | \( -\frac{147281603041}{215233605} \) | = | $-1 \cdot 3^{-16} \cdot 5^{-1} \cdot 5281^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5333229333711016770884903642$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.29086960847710152197363562428$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $Q$ | ≈ | $1.0594919023465201$ | |||
Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.366226749263402$ |
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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.2012513759725045021929325719$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $\Omega$ | ≈ | $0.20215963587963414705729524333$ | 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: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2^{2}\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)
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Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ | 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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Special value: | $ L'(E,1)$ | ≈ | $2.1029661685700951958666744096 $ | 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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Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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BSD formula
$\displaystyle 2.102966169 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.202160 \cdot 5.201251 \cdot 8}{2^2} \approx 2.102966169$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 2048 | 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 at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{4}^{*}$ | additive | -1 | 4 | 12 | 0 |
$3$ | $4$ | $I_{16}^{*}$ | additive | -1 | 2 | 22 | 16 |
$5$ | $1$ | $I_{1}$ | nonsplit 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 | 32.96.0.67 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 480 = 2^{5} \cdot 3 \cdot 5 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 319 & 448 \\ 304 & 447 \end{array}\right),\left(\begin{array}{rr} 449 & 32 \\ 448 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 318 & 395 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 421 & 32 \\ 436 & 33 \end{array}\right),\left(\begin{array}{rr} 214 & 3 \\ 365 & 268 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 420 & 449 \\ 241 & 156 \end{array}\right)$.
The torsion field $K:=\Q(E[480])$ is a degree-$11796480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/480\Z)$.
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 |
---|---|---|---|
$2$ | additive | $2$ | \( 45 = 3^{2} \cdot 5 \) |
$3$ | additive | $8$ | \( 80 = 2^{4} \cdot 5 \) |
$5$ | nonsplit multiplicative | $6$ | \( 144 = 2^{4} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 8 and 16.
Its isogeny class 720h
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 15a6, its twist by $12$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-5}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$2$ | \(\Q(\sqrt{15}) \) | \(\Z/4\Z\) | 2.2.60.1-15.1-d2 |
$2$ | \(\Q(\sqrt{-3}) \) | \(\Z/4\Z\) | 2.0.3.1-19200.1-g1 |
$4$ | \(\Q(\sqrt{-3}, \sqrt{-5})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$4$ | \(\Q(\sqrt{-2}, \sqrt{-3})\) | \(\Z/8\Z\) | not in database |
$4$ | \(\Q(\sqrt{-3}, \sqrt{10})\) | \(\Z/8\Z\) | not in database |
$8$ | 8.0.82944000000.32 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$8$ | 8.4.5184000000.1 | \(\Z/8\Z\) | not in database |
$8$ | 8.0.3317760000.8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$8$ | 8.0.5308416000000.5 | \(\Z/16\Z\) | not in database |
$8$ | 8.0.5308416000000.2 | \(\Z/16\Z\) | not in database |
$8$ | 8.2.28343520000.6 | \(\Z/6\Z\) | not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
$16$ | 16.0.26873856000000000000.6 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$16$ | deg 16 | \(\Z/16\Z\) | not in database |
$16$ | deg 16 | \(\Z/16\Z\) | not in database |
$16$ | 16.0.450868486864896000000000000.14 | \(\Z/2\Z \oplus \Z/16\Z\) | 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 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | add | add | nonsplit | ss | ord | ord | ord | ord | ss | ord | ss | ord | ord | ord | ord |
$\lambda$-invariant(s) | - | - | 1 | 1,3 | 1 | 1 | 1 | 1 | 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.