Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-1663x-25680\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-1663xz^2-25680z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-26603x-1670106\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(75, 479)$ | $4.5879433756730922486890682491$ | $\infty$ |
Integral points
\( \left(75, 479\right) \), \( \left(75, -555\right) \)
Invariants
| Conductor: | $N$ | = | \( 1369 \) | = | $37^{2}$ |
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| Discriminant: | $\Delta$ | = | $-1369$ | = | $-1 \cdot 37^{2} $ |
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| j-invariant: | $j$ | = | \( -371323264041 \) | = | $-1 \cdot 3^{3} \cdot 37 \cdot 719^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.26169607405070362892002534856$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.34012357805666711180799059661$ |
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| $abc$ quality: | $Q$ | ≈ | $1.116627416496341$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.688859661303817$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.5879433756730922486890682491$ |
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| Real period: | $\Omega$ | ≈ | $0.37382375965132293176986151217$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.7150822417614972297962238813 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.715082242 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.373824 \cdot 4.587943 \cdot 1}{1^2} \\ & \approx 1.715082242\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 252 |
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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 is only one prime $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $37$ | $1$ | $II$ | additive | 1 | 2 | 2 | 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 |
|---|---|---|
| $2$ | 2G | 4.2.0.1 |
| $7$ | 7B | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2072 = 2^{3} \cdot 7 \cdot 37 \), index $192$, genus $6$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 28 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 14 & 197 \end{array}\right),\left(\begin{array}{rr} 297 & 28 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1037 & 5 \\ 1043 & 36 \end{array}\right),\left(\begin{array}{rr} 2045 & 28 \\ 2044 & 29 \end{array}\right),\left(\begin{array}{rr} 1555 & 28 \\ 1813 & 1 \end{array}\right),\left(\begin{array}{rr} 565 & 21 \\ 1771 & 1984 \end{array}\right),\left(\begin{array}{rr} 1 & 28 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 1827 & 1919 \end{array}\right)$.
The torsion field $K:=\Q(E[2072])$ is a degree-$29388054528$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2072\Z)$.
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 |
|---|---|---|---|
| $37$ | additive | $254$ | \( 1 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 1369c
consists of 2 curves linked by isogenies of
degree 7.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.5476.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.119946304.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.1165463885299.2 | \(\Z/7\Z\) | not in database |
| $8$ | 8.2.5611243656483.1 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $18$ | 18.0.6484200109515482928287359214388780642304.1 | \(\Z/14\Z\) | 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 | ord | ss | ord | ord | ord | ord | ord | ord | ord | ord | ord | add | ord | ord | ord |
| $\lambda$-invariant(s) | ? | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | - | 1 | 1 | 1 |
| $\mu$-invariant(s) | ? | 0,0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | - | 0 | 0 | 0 |
An entry ? indicates that the invariants have not yet been computed.
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.