Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-16845x-390600\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-16845xz^2-390600z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-21831147x-18158340186\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-105, 525\right)\) |
$\hat{h}(P)$ | ≈ | $2.7060659344010859212576927062$ |
Torsion generators
\( \left(-24, 12\right) \), \( \left(140, -70\right) \)
Integral points
\( \left(-105, 525\right) \), \( \left(-105, -420\right) \), \( \left(-24, 12\right) \), \( \left(140, -70\right) \), \( \left(1985, 87260\right) \), \( \left(1985, -89245\right) \)
Invariants
Conductor: | \( 25215 \) | = | $3 \cdot 5 \cdot 41^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $240474027200625 $ | = | $3^{4} \cdot 5^{4} \cdot 41^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{111284641}{50625} \) | = | $3^{-4} \cdot 5^{-4} \cdot 13^{3} \cdot 37^{3}$ | 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);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $1.4545084612693101144897851894\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.40227757208284378744359649712\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0253374473912513\dots$ | |||
Szpiro ratio: | $4.026467856821097\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.7060659344010859212576927062\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.43747489206741805936904000270\dots$ | 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: | $ 64 $ = $ 2^{2}\cdot2^{2}\cdot2^{2} $ | 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: | $4$ | 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) $ ≈ $ 4.7353436103177274476276749634 $ | 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
$\displaystyle 4.735343610 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.437475 \cdot 2.706066 \cdot 64}{4^2} \approx 4.735343610$
Modular invariants
Modular form 25215.2.a.f
For more coefficients, see the Downloads section to the right.
Modular degree: | 66560 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$5$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$41$ | $4$ | $I_0^{*}$ | 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 |
---|---|---|
$2$ | 2Cs | 8.48.0.44 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9840 = 2^{4} \cdot 3 \cdot 5 \cdot 41 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 4839 & 7708 \\ 8036 & 1641 \end{array}\right),\left(\begin{array}{rr} 821 & 7216 \\ 656 & 657 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 8 & 129 \end{array}\right),\left(\begin{array}{rr} 6725 & 6724 \\ 8692 & 8365 \end{array}\right),\left(\begin{array}{rr} 3281 & 3608 \\ 164 & 4593 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 8399 & 0 \\ 0 & 9839 \end{array}\right),\left(\begin{array}{rr} 9825 & 16 \\ 9824 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[9840])$ is a degree-$2031353856000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9840\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 25215.f
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 15.a5, its twist by $41$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{41}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | 2.2.41.1-225.1-b4 |
$2$ | \(\Q(\sqrt{-41}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(i, \sqrt{41})\) | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{5}, \sqrt{41})\) | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.0.452121760000.1 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.0.9375196815360000.197 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.0.9375196815360000.94 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | 16.0.225009447236444891985739776.1 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | ord | split | split | ss | ord | ord | ord | ord | ss | ord | ss | ord | add | ord | ord |
$\lambda$-invariant(s) | 4 | 4 | 2 | 1,1 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1,1 | 1 | - | 1 | 1 |
$\mu$-invariant(s) | 1 | 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.