Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-36x-70\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-36xz^2-70z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-46035x-3116178\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{6}\Z\)
Mordell-Weil generators
$P$ | $\hat{h}(P)$ | Order |
---|---|---|
$(-4, 5)$ | $0$ | $6$ |
Integral points
\( \left(-4, 5\right) \), \( \left(-4, -2\right) \), \( \left(10, 19\right) \), \( \left(10, -30\right) \)
Invariants
Conductor: | $N$ | = | \( 14 \) | = | $2 \cdot 7$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $\Delta$ | = | $941192$ | = | $2^{3} \cdot 7^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | $j$ | = | \( \frac{4956477625}{941192} \) | = | $2^{-3} \cdot 5^{3} \cdot 7^{-6} \cdot 11^{3} \cdot 31^{3}$ | 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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.13620516391097038811684026362$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.13620516391097038811684026362$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $Q$ | ≈ | $1.0082122835525031$ | |||
Szpiro ratio: | $\sigma_{m}$ | ≈ | $8.45906639413832$ |
BSD invariants
Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Mordell-Weil rank: | $r$ | = | $ 0$ | comment: Rank
sage: E.rank()
gp: [lower,upper] = ellrank(E)
magma: Rank(E);
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Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $\Omega$ | ≈ | $1.9813419560668832341695716767$ | 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: | $\prod_{p}c_p$ | = | $ 6 $ = $ 1\cdot( 2 \cdot 3 ) $ | 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: | $\#E(\Q)_{\mathrm{tor}}$ | = | $6$ | 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)$ | ≈ | $0.33022365934448053902826194612 $ | 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${}_{\mathrm{an}}$ | = | $1$ (exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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BSD formula
$\displaystyle 0.330223659 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.981342 \cdot 1.000000 \cdot 6}{6^2} \approx 0.330223659$
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: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data at primes of bad reduction
This elliptic curve is semistable. There are 2 primes $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
$7$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
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 | 8.6.0.6 |
$3$ | 3Cs.1.1 | 3.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 325 & 276 \\ 18 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 469 & 36 \\ 468 & 37 \end{array}\right),\left(\begin{array}{rr} 445 & 36 \\ 174 & 277 \end{array}\right),\left(\begin{array}{rr} 253 & 36 \\ 0 & 29 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 10 & 3 \\ 249 & 352 \end{array}\right),\left(\begin{array}{rr} 19 & 24 \\ 432 & 307 \end{array}\right)$.
The torsion field $K:=\Q(E[504])$ is a degree-$13934592$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/504\Z)$.
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 |
---|---|---|---|
$2$ | nonsplit multiplicative | $4$ | \( 1 \) |
$3$ | good | $2$ | \( 1 \) |
$7$ | split multiplicative | $8$ | \( 2 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 14.a
consists of 6 curves linked by isogenies of
degrees dividing 18.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | 2.2.8.1-98.1-a5 |
$2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z \oplus \Z/6\Z\) | 2.0.3.1-196.2-a4 |
$4$ | 4.0.1568.1 | \(\Z/12\Z\) | not in database |
$4$ | \(\Q(\sqrt{2}, \sqrt{-3})\) | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
$8$ | 8.4.205520896.2 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
$8$ | 8.0.157351936.3 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
$8$ | 8.0.199148544.2 | \(\Z/3\Z \oplus \Z/12\Z\) | not in database |
$9$ | 9.3.2315685267.2 | \(\Z/18\Z\) | not in database |
$16$ | deg 16 | \(\Z/24\Z\) | not in database |
$16$ | deg 16 | \(\Z/6\Z \oplus \Z/12\Z\) | not in database |
$16$ | 16.0.162447943996702457856.4 | \(\Z/6\Z \oplus \Z/12\Z\) | not in database |
$18$ | 18.0.3674701490222020866048.1 | \(\Z/3\Z \oplus \Z/18\Z\) | not in database |
$18$ | 18.0.432324955623130532869681152.2 | \(\Z/3\Z \oplus \Z/18\Z\) | not in database |
$18$ | 18.0.144784752906623254803.2 | \(\Z/3\Z \oplus \Z/18\Z\) | not in database |
$18$ | 18.6.719728910524754422652731392.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | 7 |
---|---|---|---|
Reduction type | nonsplit | ord | split |
$\lambda$-invariant(s) | 0 | 0 | 1 |
$\mu$-invariant(s) | 0 | 1 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.