Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-20691x+1078110\)
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(homogenize, simplify) |
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\(y^2z=x^3-20691xz^2+1078110z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-20691x+1078110\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(3, 1008)$ | $1.8274193988403674619833614041$ | $\infty$ |
| $(66, 0)$ | $0$ | $2$ |
| $(99, 0)$ | $0$ | $2$ |
Integral points
\( \left(-165, 0\right) \), \((3,\pm 1008)\), \( \left(66, 0\right) \), \( \left(99, 0\right) \), \((187,\pm 1936)\), \((198,\pm 2178)\), \((139491,\pm 52097760)\)
Invariants
| Conductor: | $N$ | = | \( 60984 \) | = | $2^{3} \cdot 3^{2} \cdot 7 \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $64800696812544$ | = | $2^{10} \cdot 3^{6} \cdot 7^{2} \cdot 11^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{740772}{49} \) | = | $2^{2} \cdot 3^{3} \cdot 7^{-2} \cdot 19^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3996473636546799746734348097$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.92622906754518123423618636563$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0653400939100561$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.7597192069911505$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.8274193988403674619833614041$ |
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| Real period: | $\Omega$ | ≈ | $0.60895728221565102429673879156$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2\cdot2^{2}\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.4512814023439559468411278786 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.451281402 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.608957 \cdot 1.827419 \cdot 64}{4^2} \\ & \approx 4.451281402\end{aligned}$$
Modular invariants
Modular form 60984.2.a.p
For more coefficients, see the Downloads section to the right.
| Modular degree: | 163840 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | additive | -1 | 3 | 10 | 0 |
| $3$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $11$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
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 | 8.12.0.3 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1845 & 4 \\ 1844 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 925 & 1122 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1585 & 396 \\ 594 & 793 \end{array}\right),\left(\begin{array}{rr} 463 & 396 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1175 & 0 \\ 0 & 1847 \end{array}\right),\left(\begin{array}{rr} 1231 & 0 \\ 0 & 1847 \end{array}\right)$.
The torsion field $K:=\Q(E[1848])$ is a degree-$40874803200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1848\Z)$.
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 | $2$ | \( 1089 = 3^{2} \cdot 11^{2} \) |
| $3$ | additive | $6$ | \( 6776 = 2^{3} \cdot 7 \cdot 11^{2} \) |
| $7$ | split multiplicative | $8$ | \( 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} \) |
| $11$ | additive | $62$ | \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 60984ch
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 56a2, its twist by $33$.
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 |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{-7}, \sqrt{-33})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{33})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{14}, \sqrt{-66})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.78724813483008.4 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | 16.0.34822159495883806555961819136.6 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | add | ord | split | add | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 1 | 4 | - | 3 | 1 | 1 | 1,1 | 1 | 1 | 1 | 3 | 1 | 1 |
| $\mu$-invariant(s) | - | - | 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.