Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+24x+98\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+24xz^2+98z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+31725x+4488750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2, 11)$ | $0.49098887703616206389206789793$ | $\infty$ |
| $(-3, 1)$ | $0$ | $2$ |
Integral points
\( \left(-3, 1\right) \), \( \left(-2, 7\right) \), \( \left(-2, -6\right) \), \( \left(2, 11\right) \), \( \left(2, -14\right) \), \( \left(7, 21\right) \), \( \left(7, -29\right) \), \( \left(47, 301\right) \), \( \left(47, -349\right) \)
Invariants
| Conductor: | $N$ | = | \( 3950 \) | = | $2 \cdot 5^{2} \cdot 79$ |
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| Discriminant: | $\Delta$ | = | $-4937500$ | = | $-1 \cdot 2^{2} \cdot 5^{6} \cdot 79 $ |
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| j-invariant: | $j$ | = | \( \frac{103823}{316} \) | = | $2^{-2} \cdot 47^{3} \cdot 79^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.032320431342309524797222304058$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.83703938755935971209760197067$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8000937907084497$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.7357702447647867$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.49098887703616206389206789793$ |
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| Real period: | $\Omega$ | ≈ | $1.7140939297782127793035531553$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.6832021074326134515658917624 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.683202107 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.714094 \cdot 0.490989 \cdot 8}{2^2} \\ & \approx 1.683202107\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 768 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $79$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
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 | 8.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 632 = 2^{3} \cdot 79 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 322 & 1 \\ 471 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 81 & 554 \\ 552 & 79 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 317 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 629 & 4 \\ 628 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[632])$ is a degree-$4921712640$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/632\Z)$.
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 | $4$ | \( 1975 = 5^{2} \cdot 79 \) |
| $5$ | additive | $14$ | \( 158 = 2 \cdot 79 \) |
| $79$ | nonsplit multiplicative | $80$ | \( 50 = 2 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 3950.a
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 158.e2, its twist by $5$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-79}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.126400.3 | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.99712207360000.12 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 79 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | ord | add | ss | ord | ord | ord | ss | ss | ord | ord | ord | ord | ord | ss | nonsplit |
| $\lambda$-invariant(s) | 4 | 3 | - | 1,1 | 1 | 1 | 1 | 1,1 | 1,1 | 1 | 1 | 1 | 1 | 3 | 1,1 | 1 |
| $\mu$-invariant(s) | 0 | 0 | - | 0,0 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.