Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2-5x+2\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z-5xz^2+2z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-6507x+199206\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{4}\Z\)
Torsion generators
\( \left(-3, 1\right) \), \( \left(2, 1\right) \)
Integral points
\( \left(-3, 1\right) \), \( \left(0, 1\right) \), \( \left(0, -2\right) \), \( \left(1, -1\right) \), \( \left(2, 1\right) \), \( \left(2, -4\right) \)
Invariants
Conductor: | \( 15 \) | = | $3 \cdot 5$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $225 $ | = | $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: | \( \frac{13997521}{225} \) | = | $3^{-2} \cdot 5^{-2} \cdot 241^{3}$ | 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: | $-0.74885116236281644215221255789\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.74885116236281644215221255789\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9622954410890624\dots$ | |||
Szpiro ratio: | $6.076102575120219\dots$ |
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: | $5.6024121693304080927207233472\dots$ | 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: | $ 4 $ = $ 2\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: | $8$ | 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) $ ≈ $ 0.35015076058315050579504520920 $ | 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
$\displaystyle 0.350150761 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 5.602412 \cdot 1.000000 \cdot 4}{8^2} \approx 0.350150761$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 2 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 2 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 2 primes $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $I_{2}$ | nonsplit 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 | 16.96.0.7 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 240 = 2^{4} \cdot 3 \cdot 5 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 161 & 16 \\ 6 & 97 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 16 \\ 4 & 193 \end{array}\right),\left(\begin{array}{rr} 61 & 8 \\ 170 & 197 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 4 & 65 \end{array}\right),\left(\begin{array}{rr} 225 & 16 \\ 224 & 17 \end{array}\right),\left(\begin{array}{rr} 15 & 8 \\ 208 & 209 \end{array}\right)$.
The torsion field $K:=\Q(E[240])$ is a degree-$737280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/240\Z)$.
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 |
---|---|---|---|
$2$ | good | $2$ | \( 1 \) |
$3$ | nonsplit multiplicative | $4$ | \( 5 \) |
$5$ | split multiplicative | $6$ | \( 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 15a
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{4}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{5}) \) | \(\Z/2\Z \oplus \Z/8\Z\) | 2.2.5.1-45.1-a6 |
$4$ | \(\Q(i, \sqrt{15})\) | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
$4$ | \(\Q(\zeta_{12})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$4$ | \(\Q(\zeta_{15})^+\) | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
$8$ | 8.0.12960000.1 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
$8$ | \(\Q(\zeta_{20})\) | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
$8$ | 8.2.110716875.2 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
$16$ | 16.0.1426576071720960000.2 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
$16$ | \(\Q(\zeta_{60})\) | \(\Z/4\Z \oplus \Z/16\Z\) | not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/32\Z\) | not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | 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 |
---|---|---|---|
Reduction type | ord | nonsplit | split |
$\lambda$-invariant(s) | 0 | 0 | 1 |
$\mu$-invariant(s) | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.