Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-145x+975\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-145xz^2+975z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-11772x+746064\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-5, 40\right) \) | $0.13781230466972081831568034940$ | $\infty$ |
| \( \left(-15, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-5:40:1]\) | $0.13781230466972081831568034940$ | $\infty$ |
| \([-15:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-42, 1080\right) \) | $0.13781230466972081831568034940$ | $\infty$ |
| \( \left(-132, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-15, 0\right) \), \((-5,\pm 40)\), \((3,\pm 24)\), \((5,\pm 20)\), \((10,\pm 25)\), \((35,\pm 200)\), \((65,\pm 520)\), \((475,\pm 10360)\)
\([-15:0:1]\), \([-5:\pm 40:1]\), \([3:\pm 24:1]\), \([5:\pm 20:1]\), \([10:\pm 25:1]\), \([35:\pm 200:1]\), \([65:\pm 520:1]\), \([475:\pm 10360:1]\)
\( \left(-15, 0\right) \), \((-5,\pm 40)\), \((3,\pm 24)\), \((5,\pm 20)\), \((10,\pm 25)\), \((35,\pm 200)\), \((65,\pm 520)\), \((475,\pm 10360)\)
Invariants
| Conductor: | $N$ | = | \( 320 \) | = | $2^{6} \cdot 5$ |
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| Minimal Discriminant: | $\Delta$ | = | $-256000000$ | = | $-1 \cdot 2^{14} \cdot 5^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{20720464}{15625} \) | = | $-1 \cdot 2^{4} \cdot 5^{-6} \cdot 109^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $0.31250322937848018394082865934$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.49616848127478934371260881569$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9589363178944194$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.747470091616141$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.13781230466972081831568034940$ |
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| Real period: | $\Omega$ | ≈ | $1.6080776337804031449674895669$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 2^{2}\cdot( 2 \cdot 3 ) $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.3296773087946519376042216693 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.329677309 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.608078 \cdot 0.137812 \cdot 24}{2^2} \\ & \approx 1.329677309\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 96 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}^{*}$ | additive | -1 | 6 | 14 | 0 |
| $5$ | $6$ | $I_{6}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 4.12.0.12 | $12$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 120 = 2^{3} \cdot 3 \cdot 5 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 97 & 24 \\ 96 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 13 & 24 \\ 12 & 13 \end{array}\right),\left(\begin{array}{rr} 15 & 16 \\ 74 & 95 \end{array}\right),\left(\begin{array}{rr} 59 & 96 \\ 114 & 95 \end{array}\right),\left(\begin{array}{rr} 41 & 24 \\ 100 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 116 \\ 49 & 55 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 22 \\ 14 & 83 \end{array}\right)$.
The torsion field $K:=\Q(E[120])$ is a degree-$92160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 1 \) |
| $3$ | good | $2$ | \( 64 = 2^{6} \) |
| $5$ | split multiplicative | $6$ | \( 64 = 2^{6} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 320.a
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 20.a2, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.0.4.1-6400.2-e5 |
| $2$ | \(\Q(\sqrt{6}) \) | \(\Z/6\Z\) | 2.2.24.1-100.1-b1 |
| $4$ | 4.2.1600.1 | \(\Z/4\Z\) | not in database |
| $4$ | 4.0.1280.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{6})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.1492992.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.4.3317760000.1 | \(\Z/12\Z\) | not in database |
| $8$ | 8.0.40960000.2 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.530841600.2 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $12$ | 12.0.20061226008576.4 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.35664401793024.4 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | 16.4.268435456000000000000.4 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.0.176120502681600000000.6 | \(\Z/4\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.15992037016835457024000000000000.2 | \(\Z/18\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ord | split | ord | ss | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 3 | 2 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 |
| $\mu$-invariant(s) | - | 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.