Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-x^2-25x+25\) | (homogenize, simplify) |
\(y^2z=x^3-x^2z-25xz^2+25z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-2052x+12096\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(0, 5\right)\) |
$\hat{h}(P)$ | ≈ | $0.88630155122291897351275285731$ |
Torsion generators
\( \left(1, 0\right) \), \( \left(5, 0\right) \)
Integral points
\( \left(-5, 0\right) \), \((-3,\pm 8)\), \((0,\pm 5)\), \( \left(1, 0\right) \), \( \left(5, 0\right) \), \((7,\pm 12)\), \((25,\pm 120)\), \((10087,\pm 1013028)\)
Invariants
Conductor: | \( 960 \) | = | $2^{6} \cdot 3 \cdot 5$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $921600 $ | = | $2^{12} \cdot 3^{2} \cdot 5^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{438976}{225} \) | = | $2^{6} \cdot 3^{-2} \cdot 5^{-2} \cdot 19^{3}$ | 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: | $-0.16326786508309690429818265780\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.85641504564304221371541477926\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0653400939100561\dots$ | |||
Szpiro ratio: | $3.103272640757608\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.88630155122291897351275285731\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $2.4660298168785327590088095061\dots$ | 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: | $ 16 $ = $ 2^{2}\cdot2\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: | $4$ | 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) $ ≈ $ 2.1856460520614143983442215111 $ | 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
$\displaystyle 2.185646052 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 2.466030 \cdot 0.886302 \cdot 16}{4^2} \approx 2.185646052$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 128 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{2}^{*}$ | Additive | -1 | 6 | 12 | 0 |
$3$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$5$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
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 | 4.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 120 = 2^{3} \cdot 3 \cdot 5 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 3 & 2 \\ 38 & 119 \end{array}\right),\left(\begin{array}{rr} 89 & 116 \\ 58 & 111 \end{array}\right),\left(\begin{array}{rr} 117 & 4 \\ 116 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 97 & 4 \\ 74 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 59 & 118 \\ 0 & 119 \end{array}\right)$.
The torsion field $K:=\Q(E[120])$ is a degree-$737280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 960.f
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 480.b3, its twist by $-8$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{10}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(i, \sqrt{6})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-6}, \sqrt{-10})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.3317760000.5 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.4.9437184000000.9 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.2.29023764480000.18 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | nonsplit | split | ss | ord | ord | ord | ord | ord | ord | ss | ord | ord | ord | ord |
$\lambda$-invariant(s) | - | 3 | 2 | 1,1 | 1 | 1 | 1 | 1 | 3 | 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.