Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-x^2-36560x+8817600\) | (homogenize, simplify) |
\(y^2z=x^3-x^2z-36560xz^2+8817600z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-2961387x+6419146266\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-95, 3380\right)\) |
$\hat{h}(P)$ | ≈ | $2.2331584861859613582576342821$ |
Torsion generators
\( \left(-264, 0\right) \)
Integral points
\( \left(-264, 0\right) \), \((-95,\pm 3380)\), \((760,\pm 20480)\)
Invariants
Conductor: | \( 40560 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 13^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-30367656443904000 $ | = | $-1 \cdot 2^{24} \cdot 3 \cdot 5^{3} \cdot 13^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{273359449}{1536000} \) | = | $-1 \cdot 2^{-12} \cdot 3^{-1} \cdot 5^{-3} \cdot 11^{3} \cdot 59^{3}$ | 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: | $1.8466406639471200483159860423\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.12898119534359362912798979994\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0491971880149842\dots$ | |||
Szpiro ratio: | $4.28865167400649\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.2331584861859613582576342821\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.32128016572583940755470627456\dots$ | 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: | $ 24 $ = $ 2^{2}\cdot1\cdot3\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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 4.3048171712033419107217185782 $ | 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
$\displaystyle 4.304817171 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.321280 \cdot 2.233158 \cdot 24}{2^2} \approx 4.304817171$
Modular invariants
Modular form 40560.2.a.q
For more coefficients, see the Downloads section to the right.
Modular degree: | 331776 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{16}^{*}$ | Additive | -1 | 4 | 24 | 12 |
$3$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$5$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$13$ | $2$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 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 |
---|---|---|
$2$ | 2B | 4.6.0.1 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 352 & 1443 \\ 117 & 274 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 1537 & 24 \\ 1536 & 25 \end{array}\right),\left(\begin{array}{rr} 479 & 0 \\ 0 & 1559 \end{array}\right),\left(\begin{array}{rr} 623 & 936 \\ 1326 & 1169 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 254 & 131 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 729 & 988 \\ 260 & 989 \end{array}\right),\left(\begin{array}{rr} 456 & 1313 \\ 1391 & 1314 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$2415329280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 40560bv
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 30a3, its twist by $-52$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-15}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{39}) \) | \(\Z/12\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-65}) \) | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-15}, \sqrt{39})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$6$ | 6.0.8302761792.20 | \(\Z/6\Z\) | Not in database |
$8$ | 8.0.5205242250000.9 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.4.34113075609600.16 | \(\Z/24\Z\) | Not in database |
$8$ | 8.0.263218176000000.2 | \(\Z/8\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | Not in database |
$18$ | 18.6.26829279820234328029041586176000000000000.3 | \(\Z/36\Z\) | 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 | nonsplit | split | ord | ss | add | ord | ord | ss | ord | ord | ord | ord | ord | ss |
$\lambda$-invariant(s) | - | 1 | 2 | 1 | 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.