Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-65x-1597\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-65xz^2-1597z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5292x-1148364\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(23, 98\right)\)
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| $\hat{h}(P)$ | ≈ | $0.63777099891202476607285666328$ |
Integral points
\((17,\pm 50)\), \((23,\pm 98)\)
Invariants
| Conductor: | \( 1960 \) | = | $2^{3} \cdot 5 \cdot 7^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-1054135040 $ | = | $-1 \cdot 2^{8} \cdot 5 \cdot 7^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( -\frac{1024}{35} \) | = | $-1 \cdot 2^{10} \cdot 5^{-1} \cdot 7^{-1}$ | 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: | $0.41162385661453638659618317126\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $-1.0234293382864171389013146148\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $0.63777099891202476607285666328\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.67747177910434870698399291936\dots$ | 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: | $ 8 $ = $ 2^{2}\cdot1\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: | $1$ | 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) $ ≈ $ 3.4565748263526964948143710925 $ | 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
$\displaystyle 3.456574826 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.677472 \cdot 0.637771 \cdot 8}{1^2} \approx 3.456574826$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 768 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{1}^{*}$ | Additive | 1 | 3 | 8 | 0 |
| $5$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 70 = 2 \cdot 5 \cdot 7 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 69 & 2 \\ 68 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 31 & 2 \\ 31 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 57 & 2 \\ 57 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 69 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[70])$ is a degree-$1752$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/70\Z)$.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ord | split | add | ord | ord | ord | ord | ord | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1 | 2 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1,3 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 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.
Isogenies
This curve has no rational isogenies. Its isogeny class 1960.k consists of this curve only.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.140.1 | \(\Z/2\Z\) | Not in database |
| $6$ | 6.0.686000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | Not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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.